🧠:[<think>Okay, let's see. I need to figure out how many different ways the TV station can broadcast these 5 advertisements with the given constraints. So, first, let's parse the problem again to make sure I understand all the requirements.There are 5 different advertisements: 3 commercial ones and 2 Olympic promotional ones. The last one has to be an Olympic promotional ad. Also, the two Olympic ads can't be played consecutively. I need to find the number of different broadcasting methods that satisfy these conditions, and the options are A: 120, B: 48, C: 36, D: 18.Hmm. Let me start by thinking about the constraints step by step.First, the last advertisement must be an Olympic promotional one. So, the fifth position is fixed as one of the two Olympic ads. Let's denote the commercials as C1, C2, C3 and the Olympic ads as O1, O2. So, the fifth position is either O1 or O2. That gives two choices for the last slot.Then, the remaining four positions (positions 1 to 4) need to be filled with the remaining 4 ads: 3 commercials and 1 Olympic ad. But there's another constraint: the two Olympic ads cannot be played consecutively. Since one Olympic ad is already in the fifth position, we have to make sure that the remaining Olympic ad isn't placed in the fourth position. Because if the fourth position is the other Olympic ad, then positions 4 and 5 would both be Olympic, which is not allowed.Wait, but actually, the remaining Olympic ad could be in positions 1, 2, 3, or 4, but we need to ensure that it's not adjacent to the Olympic ad in position 5. So, if the other Olympic ad is in position 4, then positions 4 and 5 would be consecutive, which is not allowed. Therefore, the remaining Olympic ad (let's say O2 if O1 is in position 5, or vice versa) cannot be in position 4. Therefore, the remaining Olympic ad can only be in positions 1, 2, or 3.But wait, maybe I need to approach this more systematically. Let's break it down.First, since the last position must be an Olympic ad, there are 2 choices for that position: O1 or O2.Once we place one Olympic ad in the fifth position, we have the remaining 4 ads: 3 commercials (C1, C2, C3) and 1 Olympic ad (either O2 or O1, depending on which one we placed in position 5). So, we have four ads left to arrange in positions 1 to 4. However, the remaining Olympic ad cannot be placed in position 4, because that would make it adjacent to the Olympic ad in position 5. Therefore, the remaining Olympic ad must be placed in positions 1, 2, or 3.So, perhaps we can calculate this as follows:1. Choose the Olympic ad for the last position: 2 choices.2. For the remaining Olympic ad (the other one), decide where to place it among positions 1, 2, or 3. That's 3 positions.3. Then, arrange the 3 commercials in the remaining 3 positions (since one position is taken by the remaining Olympic ad, and the other three positions are commercials).But wait, positions 1 to 4 would have 1 Olympic and 3 commercials. So, once we choose the position for the remaining Olympic ad (among positions 1, 2, 3), the commercials can be arranged in the remaining 3 positions in 3! ways.So, putting it all together:Number of ways = 2 (choices for last position) * 3 (positions for the remaining Olympic ad) * 3! (arrangements of commercials) = 2 * 3 * 6 = 36.But wait, 3! is 6, so 2*3*6=36. That would be option C: 36. But let me check again because sometimes these problems have subtleties.Wait, perhaps there's another way to approach this. Let's think of arranging all the commercials first and then placing the Olympic ads in the allowable slots.But since there are two Olympic ads, one of which is fixed at the end, maybe we can model this as arranging the 3 commercials and 1 Olympic ad in the first four positions, with the Olympic ad not in the fourth position. Then, place the second Olympic ad at the end.Wait, that might be another way to think about it. Let me try:First, fix the last position as an Olympic ad: 2 choices.Now, we have 4 positions left (1-4) to place 3 commercials and 1 Olympic ad. But we cannot have the Olympic ad in position 4, because that would make it adjacent to the Olympic ad in position 5. So, the Olympic ad must be in positions 1, 2, or 3.So, the number of ways to arrange positions 1-4 is: choose a position for the Olympic ad (3 choices), then arrange the 3 commercials in the remaining 3 positions (3! ways). Then multiply by the 2 choices for the last position.So again, total arrangements: 2 * 3 * 6 = 36. So, answer C: 36. That seems right. But let me make sure there's no overcounting or undercounting.Alternatively, maybe another approach: total arrangements without any constraints except the last position being Olympic. Then subtract the ones where the two Olympic ads are consecutive.Wait, but maybe that's more complicated, but let's try.Total arrangements where the last ad is Olympic: first, choose the last ad (2 choices). Then arrange the remaining 4 ads (3 commercials and 1 Olympic) in the first four positions. The number of ways is 2 * 4! / 1! (since there are 4 distinct ads). Wait, no, actually the remaining 4 ads are 3 commercials (which are distinct) and 1 Olympic ad (which is distinct as well). So, they are all distinct. So, the number of ways to arrange them is 4! = 24. So total arrangements without considering the consecutive Olympic ads: 2 * 24 = 48. Then, subtract the number of arrangements where the Olympic ad is in position 4 (since that would make it adjacent to position 5). How many such arrangements are there?If the Olympic ad is in position 4, then the last position is also Olympic. So, positions 4 and 5 are both Olympic. That's invalid. So, how many arrangements have Olympic ads in positions 4 and 5?First, fix position 5 as Olympic: 2 choices. Then, position 4 must be the other Olympic ad. Then, the remaining 3 positions (1-3) are filled with the 3 commercials. The number of ways is 2 (for the Olympic ads in positions 4 and 5) * 3! (arrangements of commercials in positions 1-3) = 2 * 6 = 12.Therefore, the total valid arrangements would be total arrangements without considering the adjacency (48) minus invalid ones (12) = 36. So, same answer: 36. So, option C. That seems to confirm the previous result.Therefore, the answer should be C: 36.Wait, but let me check again with another method. Maybe using permutations with restrictions.We have 5 positions, with the fifth being an Olympic ad. Let's denote the Olympic ads as O1 and O2. So, first, we choose which Olympic ad is last: 2 options.Then, we need to place the other Olympic ad in the first four positions such that it's not in position 4.So, the remaining Olympic ad (let's say it's O2 if the last is O1, or O1 if the last is O2) has to go into positions 1, 2, or 3. So, 3 choices.Then, the commercials are arranged in the remaining three positions. Since there are three distinct commercials, the number of arrangements is 3! = 6.Therefore, total arrangements per Olympic ad in last position: 3 * 6 = 18. Then multiply by 2 (for which Olympic ad is last): 18 * 2 = 36. Yep, same answer.Therefore, I think the correct answer is C: 36.Final Answerboxed{C}</think>The problem requires determining the number of ways to broadcast 5 advertisements (3 commercial and 2 Olympic) with the constraints that the last advertisement must be an Olympic promotional ad and the two Olympic ads cannot be broadcast consecutively.1. Fix the last position as an Olympic ad: There are 2 choices (either O1 or O2).2. Arrange the remaining 4 ads (3 commercials and 1 Olympic ad) in the first four positions: - The remaining Olympic ad cannot be placed in position 4 (to avoid consecutive Olympic ads). Therefore, it can be placed in positions 1, 2, or 3, giving 3 choices. - The 3 commercials can be arranged in the remaining 3 positions in (3!) (which is 6) ways.Calculating the total number of arrangements:- Number of choices for the last position: 2- Number of ways to place the remaining Olympic ad: 3- Number of ways to arrange the commercials: (3! = 6)Total number of broadcasting methods:[2 times 3 times 6 = 36]Thus, the number of different broadcasting methods is (boxed{C}).,<think>Okay, so I need to figure out how many different ways the TV station can broadcast these 5 advertisements given the constraints. Let me start by understanding the problem again.There are 5 advertisements in total: 3 commercial ads (let's call them C1, C2, C3) and 2 Olympic promotional ads (O1, O2). The requirements are:1. The last advertisement must be an Olympic promotional ad (either O1 or O2).2. The two Olympic promotional ads cannot be broadcast consecutively.So, I need to count all possible permutations of these 5 ads that satisfy both conditions.First, since the last ad has to be an Olympic one, the fifth position is fixed as either O1 or O2. So, there are 2 choices for the last slot. That means whatever arrangement we have for the first four positions, we need to multiply by 2 at the end.But the other condition is that the two Olympic ads can't be consecutive. Since we have to place two Olympic ads, and one is already at the end, we need to make sure that the other Olympic ad isn't in the fourth position. Because if the other Olympic ad is in the fourth position, then positions 4 and 5 would both be Olympic ads, which is not allowed.So, breaking this down:1. The fifth position is fixed as O1 or O2 (2 choices).2. The first four positions must contain the remaining 4 ads: 3 commercials (C1, C2, C3) and 1 Olympic ad (either O2 or O1, depending on which one is used in the fifth position).3. But in these first four positions, we can't have the remaining Olympic ad in the fourth position (since that would make it consecutive with the fifth position's Olympic ad). Wait, actually, if we fix the fifth position as Olympic, then the remaining Olympic ad (the other one) must be placed somewhere in the first four positions but not in the fourth position. Because if it's in the fourth position, then positions 4 and 5 would both be Olympic, which is not allowed.So, the problem becomes: how many ways can we arrange 3 commercials and 1 Olympic ad in the first four positions, with the Olympic ad not in the fourth position, and then multiply by 2 (for the two choices of Olympic ads in the fifth position).Let me verify that. So, total arrangements would be:Number of ways to choose which Olympic ad is last: 2 (O1 or O2).For each of those choices, the remaining Olympic ad is fixed (either O2 or O1). Then, we need to arrange the 3 commercials and the remaining Olympic ad in the first four positions, with the Olympic ad not in the fourth position.So, first, let's handle arranging 4 ads (3 C's and 1 O) in the first four positions, with the restriction that the O cannot be in the fourth position.To calculate the number of such arrangements, we can compute the total number of arrangements without restriction and subtract the number of arrangements where O is in the fourth position.Total arrangements without restriction: Since there are 4 positions, with 3 identical commercials and 1 Olympic ad. Wait, no, wait. Wait, the commercials are different (they are 3 different commercials) and the Olympic ads are also different. Wait, in the problem statement, it says "5 different advertisements," including 3 different commercials and 2 different Olympic ads. So all ads are distinct. That's important.So, if all ads are different, then when we fix the fifth position as an Olympic ad, the remaining Olympic ad is distinct from the fifth one. Then, in the first four positions, we have 3 distinct commercials and 1 distinct Olympic ad. All of these are different.Therefore, the problem is that the first four positions must contain the three distinct commercials and one distinct Olympic ad, with the Olympic ad not in the fourth position.So, total number of ways to arrange these four distinct ads in the first four positions: 4! = 24.But we need to subtract the number of arrangements where the Olympic ad is in the fourth position.If the Olympic ad is fixed in the fourth position, then the first three positions are occupied by the three commercials. Since the commercials are distinct, the number of ways is 3! = 6.Therefore, the number of valid arrangements for the first four positions is 24 - 6 = 18.Then, since the fifth position can be filled in 2 ways (either O1 or O2), the total number of arrangements is 18 * 2 = 36.Wait, but let me check if this is correct.Alternatively, maybe another approach: first, choose which Olympic ad is last (2 choices). Then, place the other Olympic ad in one of the first three positions (since it can't be in the fourth). Then arrange the commercials in the remaining slots.So, step-by-step:1. Choose the last ad: 2 options (O1 or O2).2. Then, the other Olympic ad (let's say if we chose O1 last, then the remaining Olympic ad is O2) needs to be placed in positions 1, 2, or 3. So 3 choices.3. Then, the remaining three positions (since one Olympic is in positions 1-3 and the last is fixed) are filled with the three commercials, which are distinct. So, the number of ways to arrange the three commercials in the remaining three positions is 3! = 6.Therefore, total number of arrangements: 2 (choices for last) * 3 (positions for the other Olympic) * 6 (arrangements of commercials) = 2*3*6=36. So that gives 36, which matches the previous result.Therefore, the answer is 36, which is option C.But let me verify once again with another method to ensure that I'm not missing anything.Another approach: total number of ways to arrange the ads with the last one being Olympic, minus the number of arrangements where the two Olympic ads are consecutive and the last one is Olympic.First, total number of arrangements where the last ad is Olympic: we have two choices for the last position (O1 or O2). The first four positions are filled with the remaining 4 ads (3 commercials and 1 Olympic). Since all ads are distinct, the number of arrangements is 2 * 4! = 2 * 24 = 48.But then, we need to subtract the number of arrangements where the two Olympic ads are consecutive with the last one being Olympic. If both Olympic ads are consecutive at the end, meaning positions 4 and 5 are both Olympic. The number of such arrangements: first, choose the order of the two Olympic ads in positions 4 and 5: 2! = 2. Then, arrange the remaining 3 commercials in positions 1-3: 3! = 6. So total is 2 * 6 = 12.Therefore, total valid arrangements would be 48 - 12 = 36. So again, 36, which is option C. Therefore, this seems consistent.Alternatively, perhaps using permutations with restrictions.Since the last position must be Olympic, and the other Olympic cannot be adjacent to it. So, the other Olympic must be in positions 1, 2, or 3.Therefore, the steps:1. Assign the last position: 2 choices.2. Assign the remaining Olympic ad to positions 1-3: 3 choices.3. Assign the three commercials to the remaining three positions: 3! = 6.Total: 2 * 3 * 6 = 36. Yep, same answer.Alternatively, using the slot method.First, fix the last slot as Olympic: 2 options. Then, we have four slots left (positions 1-4) to place the remaining four ads (3 commercials and 1 Olympic). The Olympic ad cannot be in position 4 (since that would make it adjacent to position 5). So, the Olympic ad can be in positions 1, 2, or 3: 3 choices. The commercials can be arranged in the remaining three slots in 3! ways. So total: 2 * 3 * 6 = 36.All methods lead to 36. So, the answer should be C: 36.Final Answerboxed{C}</think>The TV station needs to broadcast 5 different advertisements, including 3 commercial advertisements (C1, C2, C3) and 2 Olympic promotional advertisements (O1, O2). The constraints are that the last advertisement must be an Olympic promotional advertisement, and the two Olympic promotional advertisements cannot be broadcast consecutively.1. Fix the last advertisement as an Olympic promotional ad: There are 2 choices (O1 or O2).2. Arrange the remaining 4 advertisements (3 commercials and 1 Olympic ad) in the first four positions: - The remaining Olympic ad cannot be in the fourth position (to avoid consecutive Olympic ads). - Calculate the number of valid arrangements for the first four positions: - Total arrangements without restriction: (4! = 24) - Subtract arrangements where the Olympic ad is in the fourth position: (3! = 6) - Valid arrangements: (24 - 6 = 18)3. Multiply by the number of choices for the last Olympic ad: (18 times 2 = 36)Thus, the number of different broadcasting methods is (boxed{C}).]

🔑:AnalysisAccording to the problem, to meet the conditions, we first choose one of the two Olympic advertisements to be placed in the last position, which results in C_2^1=2 outcomes.Since the two Olympic advertisements cannot be broadcast consecutively, the second Olympic advertisement can only choose one of the first three positions, resulting in 3 outcomes. The remaining three elements are fully permuted in three positions, resulting in A_3^3 outcomes. In total, there are 2 times 3 times A_3^3 = 36 outcomes.Therefore, the answer is boxed{C}.

🧠:[<think>Alright, so I have this integral to solve: the integral from 0 to π/3 of x times tan squared x dx. Hmm, okay. Let me think about how to approach this. I remember that integrals involving x times some trigonometric function often require integration by parts. But first, maybe I should recall what tan squared x is in terms of other trigonometric functions? Because I know there's an identity that relates tan squared x to sec squared x. Let me check.Yes, the identity is tan²x = sec²x - 1. That's right. So maybe I can rewrite the integral in terms of sec²x, which might be easier to integrate. Let me try that first.So, substituting tan²x with sec²x - 1, the integral becomes:∫₀^{π/3} x (sec²x - 1) dx = ∫₀^{π/3} x sec²x dx - ∫₀^{π/3} x dxOkay, so now the integral is split into two parts. The second part is straightforward: ∫x dx from 0 to π/3. That's just (1/2)x² evaluated from 0 to π/3, which would be (1/2)(π/3)² - 0 = π²/18. So that part is simple.The first integral is ∫x sec²x dx. Hmm, this seems like a good candidate for integration by parts. Remember, integration by parts formula is ∫u dv = uv - ∫v du. Let me choose u and dv here. Let me set u = x, which would mean du = dx. Then dv would be sec²x dx. What's the integral of sec²x? That's tan x, right? So v = tan x. So applying integration by parts:∫x sec²x dx = x tan x - ∫tan x dxOkay, now I need to compute ∫tan x dx. The integral of tan x is -ln |cos x| + C. Let me verify that. Yes, because d/dx (-ln |cos x|) = - ( -sin x / cos x ) = tan x. So that's correct. So ∫tan x dx = -ln |cos x| + C. Putting it all together:∫x sec²x dx = x tan x - (-ln |cos x|) + C = x tan x + ln |cos x| + CSince we're dealing with the integral from 0 to π/3, let's compute the definite integral:[x tan x + ln |cos x|] from 0 to π/3First, evaluate at π/3:At x = π/3: (π/3) tan(π/3) + ln(cos(π/3))tan(π/3) is √3, and cos(π/3) is 0.5. So:(π/3)(√3) + ln(0.5)Then evaluate at 0:At x = 0: 0 * tan(0) + ln(cos(0)) = 0 + ln(1) = 0So the definite integral from 0 to π/3 is:[(π/3)(√3) + ln(0.5)] - 0 = (π√3)/3 + ln(1/2)Note that ln(1/2) is equal to -ln 2. So this simplifies to:(π√3)/3 - ln 2Okay, so going back to the original split integral:∫₀^{π/3} x tan²x dx = [∫x sec²x dx] - [∫x dx] = [(π√3)/3 - ln 2] - [π²/18]So the entire integral is (π√3)/3 - ln 2 - π²/18Let me double-check my steps to make sure I didn't make any mistakes. First, substitution of tan²x with sec²x - 1, which is correct. Then splitting the integral into two parts. Then handling the second integral ∫x dx, which is straightforward. Then integration by parts for ∫x sec²x dx: setting u = x, dv = sec²x dx, which seems right. Calculated v = tan x, then got uv - ∫v du as x tan x - ∫tan x dx. Then integral of tan x is -ln |cos x|, so the integral becomes x tan x + ln |cos x|. Evaluated from 0 to π/3. At π/3, computed correctly: tan π/3 is √3, cos π/3 is 0.5. At 0, everything is 0 or ln 1, which is 0. So the definite integral for the first part is (π√3)/3 + ln(0.5), which is (π√3)/3 - ln 2. Then subtract the second integral, which was π²/18. So the final answer is (π√3)/3 - ln 2 - π²/18. That seems right.Wait, let me check if all the signs are correct. The integral ∫x tan²x dx becomes ∫x sec²x dx - ∫x dx. So after computing ∫x sec²x dx as (π√3)/3 - ln 2, then subtract ∫x dx which is π²/18, so the total is (π√3)/3 - ln 2 - π²/18. Yes, that's correct. The subtraction applies to both terms. So no sign errors there.Let me also verify the integration by parts again. Starting with ∫x sec²x dx, u = x, dv = sec²x dx. Then du = dx, v = tan x. So integration by parts gives uv - ∫v du = x tan x - ∫tan x dx. The integral of tan x is -ln |cos x|, so the result is x tan x + ln |cos x|. That's correct.Evaluated at π/3: x tan x = (π/3)(√3), and ln(cos π/3) = ln(1/2) = -ln 2. Then subtract the value at 0, which is 0 + ln(1) = 0. So yes, that part is correct.Therefore, combining the results:First integral (after parts): (π√3)/3 - ln 2Second integral: π²/18Therefore, the original integral is (π√3)/3 - ln 2 - π²/18.I think that's the final answer. Let me write it out clearly:π√3 / 3 - π² / 18 - ln 2Alternatively, this could be written with a common denominator or factored differently, but unless specified, this should be acceptable. Let me check if these terms can be combined or simplified further. The first term is π√3 / 3, the second is π² / 18, and the third is ln 2. They all have different terms, so I don't think they can be combined. So the answer is π√3 / 3 - π² / 18 - ln 2.Just to be thorough, let me compute numerical values to check if the answer makes sense. Let's compute each term:π ≈ 3.1416√3 ≈ 1.732π√3 ≈ 3.1416 * 1.732 ≈ 5.441π√3 / 3 ≈ 5.441 / 3 ≈ 1.8137π² ≈ (3.1416)^2 ≈ 9.8696π² / 18 ≈ 9.8696 / 18 ≈ 0.5483ln 2 ≈ 0.6931So putting it all together:1.8137 - 0.5483 - 0.6931 ≈ 1.8137 - 1.2414 ≈ 0.5723Now, let me check the integral numerically. Let's compute the original integral ∫₀^{π/3} x tan²x dx numerically to see if it's approximately 0.5723.First, note that tan²x = sec²x - 1, so the integral is ∫x(sec²x -1 )dx. So as before. But maybe using numerical approximation. Let's compute the integral using, say, the trapezoidal rule or Simpson's rule for an approximation.Alternatively, use a calculator for definite integral. Since I can't compute it exactly here, but based on the numerical value, our analytical answer gives approximately 0.5723. Let me see if that makes sense.Alternatively, think about the behavior of the function. From 0 to π/3, x tan²x is increasing because both x and tan²x are increasing. At x=0, the integrand is 0. At x=π/3, tan(π/3)=√3, so tan²(π/3)=3, so the integrand is (π/3)*3 = π ≈ 3.1416. So the function goes from 0 up to π at the upper limit. The integral is the area under the curve from 0 to π/3, which is a positive value. Our analytical answer gave approximately 0.57, which is plausible. If I recall, the integral of x tan²x over this interval should indeed be positive and less than, say, the integral of x*3 (since tan²x ≤3 in [0, π/3]), but the integral of 3x from 0 to π/3 is (3/2)(π/3)^2 = (3/2)(π²/9) = π²/6 ≈ 1.6449, which is bigger than our result. So 0.57 is reasonable.Alternatively, if we compute the numerical integral step by step. Let's take x=0 to x=π/3≈1.0472. Let's split the interval into, say, 4 subintervals for a rough estimate.But maybe that's overkill. Since our analytical solution gives 0.5723, which seems plausible, and the steps check out, I think the answer is correct. So the final answer should be π√3 /3 - π²/18 - ln 2.Final AnswerThe value of the integral is boxed{dfrac{pi sqrt{3}}{3} - dfrac{pi^2}{18} - ln 2}.</think>To compute the integral (int_{0}^{pi / 3} x tan ^{2}(x) , dx), we start by using the trigonometric identity (tan^2 x = sec^2 x - 1). This allows us to rewrite the integral as:[int_{0}^{pi / 3} x tan ^{2}(x) , dx = int_{0}^{pi / 3} x (sec^2 x - 1) , dx = int_{0}^{pi / 3} x sec^2 x , dx - int_{0}^{pi / 3} x , dx]First, we evaluate the simpler integral (int_{0}^{pi / 3} x , dx):[int_{0}^{pi / 3} x , dx = left[ frac{1}{2} x^2 right]_{0}^{pi / 3} = frac{1}{2} left( left( frac{pi}{3} right)^2 - 0 right) = frac{pi^2}{18}]Next, we use integration by parts on (int_{0}^{pi / 3} x sec^2 x , dx). Let (u = x) and (dv = sec^2 x , dx). Then (du = dx) and (v = tan x). Applying integration by parts:[int x sec^2 x , dx = x tan x - int tan x , dx]The integral (int tan x , dx) is (-ln |cos x| + C). Therefore,[int x sec^2 x , dx = x tan x + ln |cos x| + C]Evaluating this definite integral from 0 to (pi/3):[left[ x tan x + ln |cos x| right]_{0}^{pi / 3}]At (x = pi/3):[left( frac{pi}{3} tan left( frac{pi}{3} right) + ln left( cos left( frac{pi}{3} right) right) right) = frac{pi}{3} sqrt{3} + ln left( frac{1}{2} right) = frac{pi sqrt{3}}{3} - ln 2]At (x = 0):[0 cdot tan 0 + ln (cos 0) = 0 + ln 1 = 0]Thus, the definite integral (int_{0}^{pi / 3} x sec^2 x , dx) is:[frac{pi sqrt{3}}{3} - ln 2]Combining the results of the two integrals, we get:[int_{0}^{pi / 3} x tan ^{2}(x) , dx = left( frac{pi sqrt{3}}{3} - ln 2 right) - frac{pi^2}{18}]The final answer is:[boxed{dfrac{pi sqrt{3}}{3} - dfrac{pi^2}{18} - ln 2}],<think>Okay, so I need to compute the integral of x times tan squared x from 0 to π/3. Hmm, let's see. First, I remember that integrating products like x times a trigonometric function might require integration by parts. But before jumping into that, maybe I can simplify tan squared x somehow. Wait, there's a trigonometric identity for tan squared x. Let me recall... Oh right, tan squared x is equal to sec squared x minus 1. So, maybe rewriting the integral in terms of sec squared x will make it easier. Let me write that down:∫₀^{π/3} x tan²x dx = ∫₀^{π/3} x (sec²x - 1) dx = ∫₀^{π/3} x sec²x dx - ∫₀^{π/3} x dxOkay, so now the integral is split into two parts. The second part, ∫x dx from 0 to π/3, seems straightforward. That should be (1/2)x² evaluated from 0 to π/3, which would be (1/2)(π/3)² - 0 = π²/18. So that part is easy.Now the first integral, ∫x sec²x dx. This definitely looks like a candidate for integration by parts. The formula for integration by parts is ∫u dv = uv - ∫v du. So, I need to choose u and dv here. Let me set u = x, which means du = dx. Then dv would be sec²x dx, and integrating that gives v = tan x. Because the integral of sec²x is tan x. So applying integration by parts:∫x sec²x dx = uv - ∫v du = x tan x - ∫tan x dxNow, the integral of tan x dx is a standard one. The integral of tan x is -ln |cos x| + C. So, putting that back into the equation:∫x sec²x dx = x tan x - (-ln |cos x|) + C = x tan x + ln |cos x| + CSince we're dealing with the definite integral from 0 to π/3, we need to evaluate this expression at the bounds. So, the first part of our original integral becomes:[x tan x + ln |cos x|] from 0 to π/3Let's compute this. At x = π/3:π/3 * tan(π/3) + ln(cos(π/3))tan(π/3) is √3, so π/3 * √3. cos(π/3) is 0.5, so ln(0.5) is ln(1/2) which is -ln 2. Therefore, the upper limit contributes (π√3)/3 - ln 2.At x = 0:0 * tan 0 + ln(cos 0). tan 0 is 0, so the first term is 0. cos 0 is 1, so ln 1 is 0. So the lower limit contributes 0.Therefore, the first integral, ∫x sec²x dx from 0 to π/3, is (π√3)/3 - ln 2.Now, putting it all back together. The original integral was the first part minus the second part:[ (π√3)/3 - ln 2 ] - [ π²/18 ]So, combining those terms:Result = (π√3)/3 - ln 2 - π²/18Wait, let me check my steps again to make sure I didn't make a mistake. So, the integral of x tan²x dx is split into ∫x sec²x dx minus ∫x dx. Then integration by parts on the first integral gives x tanx + ln |cosx| evaluated from 0 to π/3. Then subtract the integral of x dx, which is π²/18. Yes, that seems right.Let me verify the integration by parts again. u = x, du = dx. dv = sec²x dx, v = tan x. Then ∫x sec²x dx = x tanx - ∫tanx dx. And ∫tanx dx = -ln |cosx| + C. So yes, that becomes x tanx + ln |cosx| + C. So evaluated at π/3 and 0. Upper limit: π/3 * tan(π/3) + ln(cos π/3). Lower limit: 0 * tan0 + ln(cos0) which is 0 + ln1=0. So upper limit is π/3 * √3 + ln(0.5) = (π√3)/3 - ln2. Then subtract the second integral, which is π²/18. So the final answer is (π√3)/3 - ln2 - π²/18.I think that's correct. Let me check if all the algebraic manipulations are right. For example, tan π/3 is indeed √3. cos π/3 is 0.5, and ln(0.5) is -ln2. So that term is correct. The integral of x dx from 0 to π/3 is (1/2)(π/3)^2 = π²/18. Yes, that's right.Therefore, combining all the terms:(π√3)/3 - ln2 - π²/18I don't think I made a mistake here. Let me consider if there's another way to check this result. Maybe differentiate the antiderivative and see if we get back the original integrand.The antiderivative of x tan²x is supposed to be x tanx + ln |cosx| - (1/2)x² + C. Wait, because the original integral is ∫x tan²x dx = ∫x sec²x dx - ∫x dx. So the antiderivative is [x tanx + ln |cosx|] - (1/2)x² + C.So differentiating that:First term: derivative of x tanx is tanx + x sec²x (using product rule).Second term: derivative of ln |cosx| is -tanx.Third term: derivative of -(1/2)x² is -x.Putting it all together: tanx + x sec²x - tanx - x = x sec²x - x = x (sec²x -1) = x tan²x. Which is the original integrand. So the differentiation checks out. Therefore, the antiderivative is correct.Therefore, the definite integral from 0 to π/3 would indeed be:[ (π√3)/3 - ln2 - (1/2)(π/3)^2 ] - [0 + 0 - 0] = (π√3)/3 - ln2 - π²/18So, yes, the answer seems correct.Final AnswerThe value of the integral is boxed{dfrac{pi sqrt{3}}{3} - dfrac{pi^2}{18} - ln 2}.</think>To compute the integral (int_{0}^{pi / 3} x tan ^{2}(x) , dx), we start by using the trigonometric identity (tan^2(x) = sec^2(x) - 1). This allows us to rewrite the integral as:[int_{0}^{pi / 3} x tan ^{2}(x) , dx = int_{0}^{pi / 3} x (sec^2(x) - 1) , dx = int_{0}^{pi / 3} x sec^2(x) , dx - int_{0}^{pi / 3} x , dx]First, we evaluate the integral (int_{0}^{pi / 3} x , dx):[int_{0}^{pi / 3} x , dx = left[ frac{1}{2} x^2 right]_{0}^{pi / 3} = frac{1}{2} left( frac{pi}{3} right)^2 = frac{pi^2}{18}]Next, we use integration by parts on (int_{0}^{pi / 3} x sec^2(x) , dx). Let (u = x) and (dv = sec^2(x) , dx). Then (du = dx) and (v = tan(x)). Applying integration by parts:[int x sec^2(x) , dx = x tan(x) - int tan(x) , dx]The integral of (tan(x)) is (-ln|cos(x)|), so:[int x sec^2(x) , dx = x tan(x) + ln|cos(x)| + C]Evaluating this from 0 to (pi/3):- At (x = pi/3): (frac{pi}{3} tanleft(frac{pi}{3}right) + lnleft|cosleft(frac{pi}{3}right)right| = frac{pi}{3} sqrt{3} + lnleft(frac{1}{2}right) = frac{pi sqrt{3}}{3} - ln 2)- At (x = 0): (0 cdot tan(0) + ln|cos(0)| = 0 + ln(1) = 0)Thus, the integral (int_{0}^{pi / 3} x sec^2(x) , dx) evaluates to:[left( frac{pi sqrt{3}}{3} - ln 2 right) - 0 = frac{pi sqrt{3}}{3} - ln 2]Combining both parts, we get:[int_{0}^{pi / 3} x tan ^{2}(x) , dx = left( frac{pi sqrt{3}}{3} - ln 2 right) - frac{pi^2}{18}]The final answer is:[boxed{dfrac{pi sqrt{3}}{3} - dfrac{pi^2}{18} - ln 2}]]

🔑:Answer: frac{pi sqrt{3}}{3}-frac{pi^{2}}{18}-ln 2Solution: We havebegin{aligned}int_{0}^{pi / 3} x tan ^{2}(x) d x & =int_{0}^{pi / 3} xleft(-1+frac{1}{cos ^{2}(x)}right) d x & =-left.frac{x^{2}}{2}right|_{0} ^{pi / 3}+int_{0}^{pi / 3} frac{x d x}{cos ^{2}(x)} & =-left.frac{x^{2}}{2}right|_{0} ^{pi / 3}+left(left.x tan (x)right|_{0} ^{pi / 3}-int_{0}^{pi / 3} tan (x) d xright) quadleft(u=x ; d v=frac{d x}{cos ^{2}(x)}right) & =-frac{x^{2}}{2}+x tan (x)+left.ln |cos (x)|right|_{0} ^{pi / 3}=-frac{pi^{2}}{18}+frac{pi sqrt{3}}{3}-ln (2)end{aligned}

🧠:[<think>Okay, let's see. There are 5 siblings trying to split a pile of candies. The problem states that when the two oldest share the candies, there's 1 left over. Similarly, when the three oldest share, there's also 1 left, and when all five share, again 1 left. We need to find the smallest number of candies that satisfies all these conditions. Hmm, so we need a number that leaves a remainder of 1 when divided by 2, 3, and 5. Let me break this down step by step.First, let's represent the number of candies as N. The problem states N is at least 2. The conditions given are:1. When divided by 2, the remainder is 1. So, N ≡ 1 mod 2.2. When divided by 3, the remainder is 1. So, N ≡ 1 mod 3.3. When divided by 5, the remainder is 1. So, N ≡ 1 mod 5.So, we need to find the smallest N ≥ 2 that satisfies all three congruences. This seems like a classic Chinese Remainder Theorem problem where we have multiple congruencies with the same remainder. Let me recall that the Chinese Remainder Theorem says that if the moduli are pairwise coprime, there's a unique solution modulo the product of the moduli. However, in this case, the moduli are 2, 3, and 5. Let me check if they are coprime.2 and 3 are coprime, 2 and 5 are coprime, 3 and 5 are coprime. So yes, they are pairwise coprime. Therefore, the solution would be unique modulo 2×3×5 = 30. So, the general solution would be N ≡ 1 mod 30. Therefore, the smallest such N greater than or equal to 2 is 1 + 30 = 31? Wait, but 1 mod 30 is 1, which is less than 2. So the next one would be 31. Let me check if 31 satisfies all the conditions.31 divided by 2 is 15 with a remainder of 1. Good.31 divided by 3 is 10 with a remainder of 1. Good.31 divided by 5 is 6 with a remainder of 1. Good.So 31 seems to satisfy all the conditions, and since the moduli are coprime, the next possible number would be 61, 91, etc., each 30 more. But since we need the minimum number greater than or equal to 2, 31 is the answer. But wait, let me think again. Is there a smaller number that satisfies these three conditions?Let me check numbers starting from 1 upwards.N=1: Divided by 2, remainder is 1. Divided by 3, remainder is 1. Divided by 5, remainder is 1. But the problem states there's at least 2 candies, so N=1 is invalid.N=2: 2 divided by 2 is 1 with remainder 0. Doesn't satisfy the first condition.N=3: 3 divided by 2 is 1 with remainder 1. Divided by 3, remainder 0. Doesn't satisfy second condition.N=4: 4 divided by 2 is 2 with remainder 0. Nope.N=5: Divided by 2, remainder 1. Divided by 3: 5/3=1 with remainder 2. Not 1. So no.N=6: Divided by 2 remainder 0.N=7: Divided by 2 remainder 1. Divided by 3: 7/3=2 remainder 1. Divided by 5: 7/5=1 remainder 2. So the third condition fails.N=8: Divided by 2 remainder 0.N=9: Divided by 2 remainder 1. Divided by 3 remainder 0. No.N=10: Divided by 2 remainder 0.N=11: Divided by 2 remainder 1. Divided by 3: 11/3=3 remainder 2. No.N=12: Divided by 2 remainder 0.N=13: Divided by 2 remainder 1. Divided by 3: 13/3=4 remainder 1. Divided by 5: 13/5=2 remainder 3. Not 1.N=14: Divided by 2 remainder 0.N=15: Divided by 2 remainder 1. 15/3=5 remainder 0. No.N=16: Divided by 2 remainder 0.N=17: Divided by 2 remainder 1. Divided by 3: 17/3=5 remainder 2. No.N=18: Divided by 2 remainder 0.N=19: Divided by 2 remainder 1. Divided by 3: 19/3=6 remainder 1. Divided by 5: 19/5=3 remainder 4. Not 1.N=20: Divided by 2 remainder 0.N=21: Divided by 2 remainder 1. Divided by 3 remainder 0. No.N=22: Divided by 2 remainder 0.N=23: Divided by 2 remainder 1. Divided by 3: 23/3=7 remainder 2. No.N=24: Divided by 2 remainder 0.N=25: Divided by 2 remainder 1. Divided by 3: 25/3=8 remainder 1. Divided by 5: 25/5=5 remainder 0. So last condition fails.N=26: Divided by 2 remainder 0.N=27: Divided by 2 remainder 1. Divided by 3 remainder 0.N=28: Divided by 2 remainder 0.N=29: Divided by 2 remainder 1. Divided by 3: 29/3=9 remainder 2. Divided by 5: 29/5=5 remainder 4. No.N=30: Divided by 2 remainder 0.N=31: Divided by 2 remainder 1. Divided by 3 remainder 1. Divided by 5 remainder 1. So all conditions met.So, yes, 31 is the smallest number greater than or equal to 2 that satisfies all three conditions. Therefore, the minimum number of candies required is 31.Alternatively, another way to approach this is to notice that N - 1 must be divisible by 2, 3, and 5. Because N ≡1 mod 2,3,5 implies that N -1 ≡0 mod 2,3,5. Therefore, N -1 is the least common multiple (LCM) of 2,3,5. Since 2,3,5 are pairwise coprime, LCM(2,3,5)=2×3×5=30. Therefore, N -1=30 => N=31.So that's another way to see it. Either way, 31 is the answer.Final AnswerThe minimum amount of candy required is boxed{31}.</think>Given a family of 5 siblings trying to split a pile of candies, we need to find the minimum number of candies ( N ) such that when divided by 2, 3, or 5, the remainder is 1 each time. The conditions can be translated into congruences:1. ( N equiv 1 pmod{2} )2. ( N equiv 1 pmod{3} )3. ( N equiv 1 pmod{5} )These congruences imply that ( N - 1 ) must be divisible by 2, 3, and 5. The least common multiple (LCM) of 2, 3, and 5 is 30. Therefore, ( N - 1 = 30 ), so ( N = 30 + 1 = 31 ).Checking the conditions:- ( 31 div 2 = 15 ) remainder 1- ( 31 div 3 = 10 ) remainder 1- ( 31 div 5 = 6 ) remainder 1All conditions are satisfied, and 31 is the smallest number greater than 2 that meets all requirements.Thus, the minimum amount of candy required is (boxed{31}).,<think>Okay, let's see. There are 5 siblings trying to split a pile of candies, which is at least 2. The problem says that when the 2 oldest share the candies, there's 1 left over. Similarly, when the 3 oldest share, there's also 1 left. And when all 5 share, again, 1 left. We need to find the smallest number of candies that satisfies all these conditions.First, let's translate these conditions into mathematical equations. If dividing by 2 leaves a remainder of 1, that means the number of candies, let's call it N, is congruent to 1 modulo 2. In other words:N ≡ 1 mod 2Similarly, for the 3 oldest siblings sharing, leaving 1 candy:N ≡ 1 mod 3And when all 5 share, also leaving 1:N ≡ 1 mod 5So, we need a number N ≥ 2 that satisfies all three congruences:1. N ≡ 1 mod 22. N ≡ 1 mod 33. N ≡ 1 mod 5Hmm, okay. These congruences all have the same remainder, 1. I remember that when a number leaves the same remainder when divided by different moduli, there's a way to find the least common multiple (LCM) of those moduli and then adjust by the remainder. Let me recall.If N ≡ a mod m and N ≡ a mod n, then N ≡ a mod LCM(m, n), provided that m and n are coprime. Wait, but in this case, the moduli are 2, 3, and 5. These are all pairwise coprime. So their LCM would be 2*3*5=30. Therefore, the solutions to N ≡ 1 mod 2, N ≡ 1 mod 3, N ≡ 1 mod 5 would be all numbers congruent to 1 mod 30. So the smallest such number is 1. But the problem says there are at least 2 candies. So the next one would be 1 + 30 = 31. Therefore, the minimal N is 31. But wait, let me verify this.Let's check if 31 satisfies all the conditions:- Divided by 2: 31 ÷ 2 = 15 with remainder 1. Correct.- Divided by 3: 31 ÷ 3 = 10 with remainder 1. Correct.- Divided by 5: 31 ÷ 5 = 6 with remainder 1. Correct.So, 31 seems to satisfy all the conditions. But let me check if there's a smaller number that also satisfies all three conditions.Wait, the problem states the pile has at least 2 candies. So numbers like 1 would be too small, but 1 is actually congruent to 1 mod everything, but since the pile must have at least 2, 1 is invalid. The next number that would satisfy N ≡ 1 mod LCM(2,3,5)=30 is 31. So that's the next one.But maybe I can check numbers manually to ensure there's no smaller number. Let's list numbers starting from 2 upwards and check the conditions:N=2:- 2 ÷ 2 = 1 rem 0. Not 1. Doesn't satisfy first condition. Reject.N=3:- 3 ÷ 2 = 1 rem 1. Good.- 3 ÷ 3 = 1 rem 0. Not 1. Reject.N=4:- 4 ÷ 2 = 2 rem 0. Reject.N=5:- 5 ÷ 2 = 2 rem 1. Good.- 5 ÷ 3 = 1 rem 2. Not 1. Reject.N=6:- 6 ÷ 2 = 3 rem 0. Reject.N=7:- 7 ÷ 2 = 3 rem 1. Good.- 7 ÷ 3 = 2 rem 1. Good.- 7 ÷ 5 = 1 rem 2. Not 1. Reject.N=8:- 8 ÷ 2 = 4 rem 0. Reject.N=9:- 9 ÷ 2 = 4 rem 1. Good.- 9 ÷ 3 = 3 rem 0. Not 1. Reject.N=10:- 10 ÷ 2 = 5 rem 0. Reject.N=11:- 11 ÷ 2 = 5 rem 1. Good.- 11 ÷ 3 = 3 rem 2. Not 1. Reject.N=12:- 12 ÷ 2 = 6 rem 0. Reject.N=13:- 13 ÷ 2 = 6 rem 1. Good.- 13 ÷ 3 = 4 rem 1. Good.- 13 ÷ 5 = 2 rem 3. Not 1. Reject.N=14:- 14 ÷ 2 = 7 rem 0. Reject.N=15:- 15 ÷ 2 = 7 rem 1. Good.- 15 ÷ 3 = 5 rem 0. Not 1. Reject.N=16:- 16 ÷ 2 = 8 rem 0. Reject.N=17:- 17 ÷ 2 = 8 rem 1. Good.- 17 ÷ 3 = 5 rem 2. Not 1. Reject.N=18:- 18 ÷ 2 = 9 rem 0. Reject.N=19:- 19 ÷ 2 = 9 rem 1. Good.- 19 ÷ 3 = 6 rem 1. Good.- 19 ÷ 5 = 3 rem 4. Not 1. Reject.N=20:- 20 ÷ 2 = 10 rem 0. Reject.N=21:- 21 ÷ 2 = 10 rem 1. Good.- 21 ÷ 3 = 7 rem 0. Not 1. Reject.N=22:- 22 ÷ 2 = 11 rem 0. Reject.N=23:- 23 ÷ 2 = 11 rem 1. Good.- 23 ÷ 3 = 7 rem 2. Not 1. Reject.N=24:- 24 ÷ 2 = 12 rem 0. Reject.N=25:- 25 ÷ 2 = 12 rem 1. Good.- 25 ÷ 3 = 8 rem 1. Good.- 25 ÷ 5 = 5 rem 0. Not 1. Reject.N=26:- 26 ÷ 2 = 13 rem 0. Reject.N=27:- 27 ÷ 2 = 13 rem 1. Good.- 27 ÷ 3 = 9 rem 0. Not 1. Reject.N=28:- 28 ÷ 2 = 14 rem 0. Reject.N=29:- 29 ÷ 2 = 14 rem 1. Good.- 29 ÷ 3 = 9 rem 2. Not 1. Reject.N=30:- 30 ÷ 2 = 15 rem 0. Reject.N=31:- 31 ÷ 2 = 15 rem 1. Good.- 31 ÷ 3 = 10 rem 1. Good.- 31 ÷ 5 = 6 rem 1. Good.So, 31 is indeed the first number that satisfies all three conditions. Therefore, the minimal number of candies required is 31.Alternatively, using the Chinese Remainder Theorem approach, since all moduli (2, 3, 5) are coprime, the solution is unique modulo 2*3*5=30. Since N ≡ 1 mod 2, 3, 5, then N ≡1 mod 30, so the minimal N is 1, but since we need at least 2, the next one is 1 + 30 =31. So, that's another way to see it.Therefore, 31 is the answer.Final AnswerThe minimum number of candies required is boxed{31}.</think>Given a family of 5 siblings trying to split a pile of candies with the following conditions:1. When the 2 oldest siblings share the candies, there is 1 left over.2. When the 3 oldest siblings share the candies, there is 1 left over.3. When all 5 siblings share the candies, there is 1 left over.We need to find the minimum number of candies ( N ) that satisfies these conditions. Translating the conditions into congruences:1. ( N equiv 1 pmod{2} )2. ( N equiv 1 pmod{3} )3. ( N equiv 1 pmod{5} )Since 2, 3, and 5 are pairwise coprime, the solution to these congruences can be found using the Chinese Remainder Theorem. The solution is ( N equiv 1 pmod{text{lcm}(2, 3, 5)} ). The least common multiple (LCM) of 2, 3, and 5 is 30. Therefore, the smallest ( N ) that satisfies all conditions is ( 1 mod 30 ). However, since the pile must have at least 2 candies, the next number is ( 1 + 30 = 31 ).Verifying:- ( 31 div 2 = 15 ) remainder 1- ( 31 div 3 = 10 ) remainder 1- ( 31 div 5 = 6 ) remainder 1Thus, the minimum number of candies required is (boxed{31}).]

🔑:1. We start by translating the problem statement into mathematical congruences. The problem states that: - If the 2 oldest siblings share the candy equally, they will have 1 piece left over. This can be written as: [ N equiv 1 pmod{2} ] - If the 3 oldest siblings share the candy equally, they will also have 1 piece left over. This can be written as: [ N equiv 1 pmod{3} ] - If all 5 siblings share the candy equally, they will also have 1 piece left over. This can be written as: [ N equiv 1 pmod{5} ]2. We need to find the smallest positive integer ( N ) that satisfies all three congruences simultaneously. This can be done by finding the least common multiple (LCM) of the moduli (2, 3, and 5) and then solving for ( N ).3. The least common multiple of 2, 3, and 5 is: [ text{lcm}(2, 3, 5) = 30 ]4. Therefore, the three congruences can be combined into a single congruence: [ N equiv 1 pmod{30} ]5. The smallest positive integer ( N ) that satisfies ( N equiv 1 pmod{30} ) is: [ N = 30k + 1 ] where ( k ) is an integer. For the smallest solution greater than 1, we set ( k = 1 ): [ N = 30 cdot 1 + 1 = 31 ]The final answer is (boxed{31}).

🧠:[<think>Okay, let's see. The problem is asking for the number of non-negative integers that can be represented by the sum from i=0 to 7 of 3^i * a_i, where each a_i is in {-1, 0, 1}. The options are given, so I need to figure out which one is correct. Hmm, let's start by understanding the problem better.First, each term in the sum is 3^i multiplied by either -1, 0, or 1. So each coefficient a_i can be -1, 0, or 1. That means each term can subtract 3^i, leave it out, or add it. So the entire sum is like a number in base 3, but instead of digits being 0, 1, 2, they can be -1, 0, 1. I remember that this is similar to balanced ternary notation. In balanced ternary, each digit is -1, 0, or 1, which allows representation of numbers without the need for a minus sign, as opposed to regular ternary.But here, the question is specifically about non-negative integers. So we need to count how many different non-negative numbers can be formed by such sums. The exponents go from i=0 to 7, so that's 8 digits in balanced ternary, right? From 3^0 up to 3^7. So the maximum possible value would be when all a_i are 1, which is the sum from i=0 to 7 of 3^i. That sum is (3^8 - 1)/2, because it's a geometric series: 1 + 3 + 9 + ... + 3^7 = (3^8 - 1)/2. Let me compute that: 3^8 is 6561, so (6561 -1)/2 is 6560/2 = 3280. So the maximum number is 3280.Similarly, the minimum possible value is when all a_i are -1, which would be -3280. But since we are only interested in non-negative integers, we need to count how many numbers from 0 up to 3280 can be represented in this balanced ternary system with 8 digits.But wait, balanced ternary can represent both positive and negative numbers. However, each number has a unique representation in balanced ternary. So for each number between -3280 and 3280, there's a unique way to represent it using 8 balanced ternary digits. But since we're only interested in non-negative numbers (0 to 3280), the number of such numbers should be 3280 + 1 = 3281. But 3281 isn't one of the options. The closest option is (D) 3381. Hmm, that's not matching. Wait, maybe my reasoning is flawed here.Wait, let's step back. If each digit can be -1, 0, 1, and there are 8 digits, then the total number of possible combinations is 3^8 = 6561. Each combination corresponds to a unique number from -3280 to 3280. However, balanced ternary is a unique representation, so each number in that range has exactly one representation. Therefore, the total numbers representable are 6561, but since each number is unique, the count from -3280 to 3280 is 6561 numbers. But how many non-negative integers are there? That would be from 0 to 3280 inclusive, which is 3281 numbers. But 3281 is not one of the options either. The options given are (A) 512, (B)729, (C)1094, (D)3381, (E)59048.Hmm, so 3281 is close to option D, which is 3381. The difference is 100. So maybe my assumption that each number from -3280 to 3280 is uniquely representable is incorrect? Wait, maybe with 8 digits in balanced ternary, the range is actually larger? Let me check.Wait, in regular balanced ternary with n digits, the range is from -((3^n -1)/2) to ((3^n -1)/2). For n=8, that would be -(6561 -1)/2 = -3280 to 3280. So that's correct. So there are 6561 numbers, each from -3280 to 3280. Therefore, non-negative numbers would be 0 to 3280, which is 3281 numbers. But 3281 is not an option. Wait, maybe the problem is that leading zeros are allowed? Wait, in balanced ternary, leading zeros don't affect the value. So for example, the number 1 can be represented as 1 followed by seven zeros, or maybe with leading zeros? Wait, no, in positional numbering systems, leading zeros don't change the value, so each number has a unique representation without leading zeros. So even if you allow leading zeros, the number itself remains the same. So the total number of distinct numbers is still 6561, with 3281 non-negative numbers.But the answer choices don't have 3281. The closest is D) 3381. So perhaps my initial reasoning is wrong. Maybe the problem isn't about unique representations, but about the total number of possible combinations, some of which might result in the same number. Wait, but the problem says "can be represented as", so maybe different combinations (different sequences of a_i) can result in the same number, and we need to count the total number of distinct non-negative integers that can be formed. So in that case, the count is not just 3281, but perhaps more? Wait, but in balanced ternary, each number has a unique representation, so different sequences of a_i would correspond to different numbers. Wait, no, balanced ternary is a unique representation system. So each number is represented by exactly one sequence of a_i. Therefore, the total number of numbers is 3^8 = 6561, but they are spread from -3280 to 3280, each number once. Therefore, non-negative numbers are 3281, which is not an option. Therefore, I must have made a mistake here.Wait, let me check the problem again. It says "non-negative integers that can be represented as sum_{i=0}^7 3^i a_i where a_i ∈ {-1,0,1}." So each a_i is in {-1,0,1}, and we need to count the number of distinct non-negative integers that can be formed by such sums. So the key point is that even though balanced ternary is a unique representation system, here, perhaps leading digits can be zero? Wait, but leading zeros don't affect the number. So maybe some numbers can be represented in multiple ways? Wait, but in balanced ternary, each number has exactly one representation. Therefore, the total number of non-negative integers would be exactly (3^8 + 1)/2. Wait, why?Because the total numbers are symmetric around zero. From -3280 to 3280, inclusive. The total count is 6561 numbers. So the number of non-negative numbers is (6561 + 1)/2 = 3281. That is, 0 to 3280 inclusive. Therefore, 3281 numbers. But 3281 is not an option. The options are 512, 729, 1094, 3381, 59048. Hmm, 3381 is 100 more than 3281. So maybe there's a miscalculation here.Wait, perhaps the problem isn't using balanced ternary? Wait, no. The coefficients are -1, 0, 1. So it is balanced ternary. Maybe the maximum value is different? Let me recalculate the maximum sum. If all a_i are 1, then the sum is 3^0 + 3^1 + ... + 3^7. Which is a geometric series with ratio 3, 8 terms. The sum is (3^8 - 1)/(3 - 1) = (6561 - 1)/2 = 6560/2 = 3280. So the maximum is 3280, and the minimum is -3280. So total numbers are 6561. Therefore, non-negative numbers are 3281. But the options don't include 3281. So perhaps the problem is different?Wait, the problem says "non-negative integers", so perhaps some of the representations result in the same integer? But balanced ternary is unique, so that can't be. Wait, unless the problem is allowing different numbers of digits. Wait, the problem specifies that the sum is from i=0 to 7, so exactly 8 digits. So each number must be represented with exactly 8 digits, possibly with leading zeros. But in balanced ternary, leading zeros would just be coefficients a_i = 0 for higher i. Wait, but in positional notation, leading zeros don't affect the value. So even if you have leading zeros, the number is the same. Therefore, the number of distinct numbers is still 6561, from -3280 to 3280. Therefore, non-negative numbers are 3281. But since 3281 isn't an option, maybe the question is asking something different?Wait, looking back at the problem statement: "non-negative integers that can be represented as sum_{i=0}^7 3^i a_i (a_i ∈ {-1,0,1}, 0 <= i <= 7)". So each a_i is in {-1,0,1}, and i ranges from 0 to 7. So each term is 3^i * a_i, so the total sum can be written as a_0 + 3 a_1 + 9 a_2 + ... + 3^7 a_7. Each a_i is -1, 0, or 1. So this is exactly balanced ternary with 8 digits.Therefore, the numbers range from -3280 to +3280, inclusive. Each number in this range is uniquely representable. Therefore, the number of non-negative integers is 3280 + 1 = 3281. But none of the options is 3281. Wait, option D is 3381. So that's 100 more. Hmm.Wait, maybe I miscalculated the maximum value. Let me check again. 3^0 + 3^1 + ... + 3^7. Let's compute this step by step:3^0 = 13^1 = 3 → sum so far: 43^2 = 9 → sum: 133^3 = 27 → sum: 403^4 = 81 → sum: 1213^5 = 243 → sum: 3643^6 = 729 → sum: 10933^7 = 2187 → sum: 3280Yes, that's correct. The maximum sum is 3280. So numbers go from -3280 to 3280. So non-negative integers are 0 to 3280, which is 3281 numbers. But answer options don't have that. So perhaps the problem isn't balanced ternary? Maybe it's allowing digits beyond 8? Wait, no, the sum is up to i=7, so 8 digits. Hmm.Alternatively, maybe the problem is counting the number of non-negative coefficients combinations? But no, that would be 3^8 = 6561, which is option B. But 6561 is the total number of combinations. But the question is about the number of non-negative integers represented, not the number of representations.Wait, another thought: perhaps some combinations result in the same non-negative integer, so even though balanced ternary is unique, if you consider that some numbers might be representable in multiple ways when allowing leading zeros? Wait, but leading zeros don't change the value. So if you have two different sequences of a_i that differ only in leading zeros, they represent the same number. But in balanced ternary, leading zeros are already part of the representation. For example, the number 1 would be represented as a_0=1, and a_1 to a_7=0. If you set a_1=1 and a_0=-1, that would be 3 -1 = 2. Wait, no, each digit is separate.Wait, perhaps the problem is not using the balanced ternary system. Maybe the coefficients are not digits in a positional number system but rather just any combination. For example, maybe overlapping combinations can produce the same number. But in balanced ternary, each number is uniquely represented. So if we use coefficients as digits in balanced ternary, then each number has a unique representation, so the total number of non-negative numbers is (3^8 +1)/2 = 3281. But since that's not an option, perhaps the question is considering something else.Wait, perhaps the problem is allowing the digits a_i to be -1, 0, 1, but treating the entire expression as a simple sum, not as a positional number system. Wait, but the expression sum_{i=0}^7 3^i a_i is exactly positional. Each a_i is a coefficient for 3^i. So that's exactly balanced ternary. So in that case, each number is unique. Therefore, the total non-negative numbers are 3281, which is not an option.Hmm. The given options are 512, 729, 1094, 3381, 59048. 729 is 3^6, 9^3, but 3^8 is 6561. 1094 is approximately 3^7 (2187) divided by 2. 3381 is roughly 3^7 * 2. 59048 is 3^10 or something. Wait, perhaps the answer is 3^8, but that's 6561, which is not an option. Option B is 729 which is 3^6. Option D is 3381. Wait, maybe my initial assumption about the maximum value is wrong. Let me recast the problem.Wait, maybe the problem is not using balanced ternary, but just any combination of coefficients. So if you have coefficients in {-1,0,1} for each power of 3, then the total number of possible sums is 3^8 = 6561. But some sums will be duplicates. So the problem is asking for the number of distinct non-negative integers among these 6561 sums. So the answer is not straightforward. Because different combinations could result in the same number.Wait, but in balanced ternary, each number has a unique representation, so duplicates don't occur. Therefore, the number of distinct sums is 6561. But they range from -3280 to 3280, so 6561 numbers. Therefore, non-negative numbers are 3281. But since this is not an option, perhaps the problem is not about balanced ternary. Wait, maybe the question is about standard ternary digits (0,1,2) converted into (-1,0,1) by subtracting 1? Wait, but the question states a_i ∈ {-1,0,1}. So no, that's not the case.Alternatively, maybe the question is allowing variable length digits? Wait, no, it's fixed from i=0 to 7. Wait, maybe there's an error in the problem's options? Or maybe the problem is presented differently. Wait, let me check again the problem statement:"There are ( ) non-negative integers that can be represented as sum_{i=0}^7 3^i a_i (a_i ∈ {-1,0,1}, 0 <= i <= 7)."Options are (A) 512, (B)729, (C)1094, (D)3381, (E)59048.Given that 3^8 = 6561, which is total possible combinations, but the numbers represented are from -3280 to 3280. If non-negative numbers are 0 to 3280, that's 3281 numbers. 3281 is not an option. So maybe the problem is not balanced ternary?Wait, another approach: maybe instead of thinking of it as balanced ternary, think of each coefficient independently. For each position i, the coefficient a_i can contribute -3^i, 0, or +3^i. So the total sum can range from - (3^0 + 3^1 + ... + 3^7) to + (3^0 + 3^1 + ... + 3^7). Which we already calculated as -3280 to +3280.But how many distinct numbers are there? Since each coefficient is independent, but the contributions can overlap. For example, 3^0 *1 + 3^1*(-1) = 1 - 3 = -2, and 3^0*(-1) + 3^1*1 = -1 + 3 = 2. So different combinations can give different numbers. But in balanced ternary, each number is uniquely represented. Wait, but here, maybe the problem is considering all possible combinations, not necessarily the unique representations. Wait, but if balanced ternary is unique, then even if you consider all combinations, the numbers would be unique. Wait, no. For example, in standard decimal, if you have digits 0-9, each number has a unique representation, so the total numbers are 10^n for n digits. But if you allow digits outside the standard range, you might get overlaps. But in balanced ternary, the uniqueness comes from the digit set {-1,0,1}. So each number is uniquely represented.Therefore, the number of distinct sums should be exactly 3^8 = 6561, spread from -3280 to 3280. Therefore, non-negative numbers would be (6561 +1)/2 = 3281. But the answer is not here. Hmm.Wait, maybe the problem is not balanced ternary, but something else. Let's consider an alternative approach. Let me think of this as a mathematical problem where each term can be -3^i, 0, or +3^i. So the total number of possible sums is the number of subsets of the set { ±3^0, ±3^1, ..., ±3^7 } where you pick at most one of each pair {3^i, -3^i}. So this is similar to subset sum problem where for each element, you can choose to add it, subtract it, or do nothing.But how does this affect the total number of possible sums? For standard subset sum where you can choose to include or exclude each element, the number of subsets is 2^n, but sums can overlap. Here, for each element, you have three choices: add, subtract, or exclude. So the number of possible combinations is 3^8 = 6561, but the number of distinct sums is less due to overlapping.Wait, but in balanced ternary, the representation is unique, so each sum corresponds to a unique number, so the number of distinct sums is 6561. But that contradicts the earlier idea that numbers range from -3280 to 3280, which is only 6561 numbers. Wait, but -3280 to 3280 inclusive is exactly 6561 numbers. Let's check: 3280 - (-3280) +1 = 6561. Yes. So each number in that interval is representable, and each combination corresponds to exactly one number. Therefore, the number of non-negative integers is 3281. But the answer isn't there. The options are (D) 3381. Hmm.Wait, maybe the problem counts zero as both positive and negative? But no, zero is non-negative. So from 0 to 3280, that's 3281 numbers. Maybe the problem has a typo, or perhaps my calculation is wrong.Wait, let's check 3^8. 3^8 is 6561. Then (3^8 -1)/2 is 3280. So the maximum positive number is 3280. Therefore, numbers from 0 to 3280 inclusive: 3281 numbers. So 3281. The answer is not there. Maybe the original problem had a different exponent? For example, if it was i=0 to 8, then 3^9 = 19683, (3^9 -1)/2 = 9841. Then non-negative numbers would be 9842, which is still not among the options. Hmm.Alternatively, maybe the question is not about balanced ternary but about a different base. Wait, but the expression is sum 3^i a_i. So base 3. If a_i can be -1,0,1, that's balanced ternary. If a_i can be 0,1,2, that's regular ternary. But here it's -1,0,1.Wait, perhaps there's a mistake in the problem's options? Or maybe in the problem statement? Alternatively, maybe the question is not about unique numbers but about something else. Wait, the problem says "non-negative integers that can be represented", so it's the count of numbers, not the count of representations. Since each number is uniquely represented, the count should be 3281. Since that's not an option, maybe the answer is D) 3381, but I need to see why.Wait, perhaps there's a miscalculation here. Let me recalculate the number of non-negative integers. If the total numbers are from -3280 to 3280 inclusive, that's 6561 numbers. Then the number of non-negative integers is 3280 +1 = 3281. So that's correct. Unless the problem is considering something else. Wait, maybe the problem allows a_i to be -1,0,1, but also allows leading digits beyond i=7? But the problem states i=0 to 7. So it's fixed at 8 digits. Therefore, leading zeros are allowed, but they don't affect the value. Therefore, the count remains 3281.Alternatively, maybe the problem is in Chinese, and there was a mistranslation. The original problem number is 18, and the options are given as (A) 512, (B)729, (C)1094, (D)3381, (E)59048. Wait, 729 is 3^6, 512 is 2^9, 1094 is approximately 3^7 + something, 3381 is around 3^7 * 4, 59048 is like 3^12 or something. Wait, maybe the problem is a different version where exponents go up to 8 instead of 7? Let's see: If the sum is up to i=8, then maximum sum is (3^9 -1)/2 = 9841. Then non-negative numbers are 9842. Still not matching.Alternatively, maybe the problem counts numbers with digits beyond 8? But the sum is up to i=7. Wait, maybe the problem is asking for numbers represented with digits a_i ∈ {0,1,2}, which is regular ternary, but the problem says a_i ∈ {-1,0,1}. So that's not the case.Alternatively, maybe the problem is considering the number of non-negative coefficients sequences instead of numbers. But no, the question is about non-negative integers.Wait, let's think differently. Maybe it's a generating function problem. The generating function for each term is x^{-3^i} + 1 + x^{3^i}. So the total generating function is the product from i=0 to 7 of (x^{-3^i} + 1 + x^{3^i}). The coefficient of x^k in this product will give the number of ways to represent k as such a sum. Therefore, the number of non-negative integers that can be represented is the number of k >= 0 for which the coefficient is at least 1.But since balanced ternary is a unique representation system, each coefficient is either 0 or 1. Wait, no, in generating functions, even with unique representations, the coefficient would be 1 for each representable number. But in reality, balanced ternary allows each number to be represented exactly once, so the generating function would indeed have coefficients 1 for each exponent from -3280 to 3280. Therefore, the number of non-negative integers is 3281. But again, not an option.Alternatively, maybe the problem is not about uniqueness. Maybe different sequences of a_i can produce the same number, hence leading to overlaps. Wait, but balanced ternary's key feature is that each number has a unique representation. So different sequences (different a_i) would produce different numbers. Therefore, the number of non-negative integers is exactly equal to the number of sequences where the sum is non-negative. But since the representations are unique, the number of non-negative integers is half of the total, rounded up. Since total is 6561, then (6561 +1)/2 = 3281. So that must be the case.Given that, and that 3281 is not an option, perhaps the problem is in base 3, but allowing digits from -1,0,1. Wait, that's balanced ternary. Wait, unless the problem is using a different base. Wait, the expression is in base 3. Each digit is multiplied by 3^i. So yes, it's base 3.Alternatively, maybe the question is about the number of numbers with absolute value less than or equal to 3280, but that's again the same as before.Alternatively, maybe the question has a typo, and the correct answer is 3281, which is close to option D, 3381. Maybe in Chinese, the numbers are similar? Or perhaps a miscalculation in the problem's creation. Alternatively, maybe I made a mistake.Wait, let me check another approach. Let's compute the total number of non-negative integers step by step for smaller n and see if a pattern emerges.For example, let's take n=1: coefficients a_0 can be -1,0,1. So possible sums: -1, 0, 1. Non-negative integers: 0,1. So 2 numbers. According to formula (3^1 +1)/2 = 2. Correct.n=2: possible sums from -4 to 4. Non-negative: 0 to 4. That's 5 numbers. Formula: (9 +1)/2=5. Correct.n=3: sums from -13 to 13. Non-negative: 14 numbers. (27 +1)/2=14. Yes.So in general, for n digits, the number of non-negative integers is (3^n +1)/2. For n=8: (6561 +1)/2=3281. So formula gives 3281. So the answer should be 3281, which is not listed. But option D is 3381. Wait, unless the question is from i=0 to 8, which would be 9 digits. Then (3^9 +1)/2=(19683 +1)/2=9842. Still not matching. Hmm.Alternatively, if the problem had i from 0 to 6 (7 digits), then (3^7 +1)/2=(2187 +1)/2=1094, which is option C. But the problem states i=0 to 7 (8 digits). So maybe the problem had a typo, and the answer is C) 1094. But the original problem says i=0 to 7. Unless there's a mistranslation or misnumbering.Alternatively, maybe the question is asking for numbers strictly greater than zero, but that would be 3280 numbers, which is still not an option. Wait, 3280 is close to option D, 3381. But not exactly.Alternatively, maybe there's a different interpretation. For each coefficient a_i, you can choose -1,0,1, but you can't have leading -1's. Wait, but in balanced ternary, leading zeros are allowed. For example, the number 1 is represented as 1 followed by seven zeros. So leading zeros don't affect the value.Alternatively, maybe the problem is allowing variable length representations, so numbers can be represented with fewer digits by omitting leading zeros, but since the problem specifies that the sum is up to i=7, the length is fixed at 8 digits. Therefore, leading zeros are required.Wait, another thought: maybe the problem isn't considering that numbers must be integers. But all the sums are integers because 3^i are integers and a_i are integers. So all sums are integers. So non-negative integers would be 0 to 3280 inclusive, which is 3281 numbers. So still not matching.Alternatively, perhaps the problem is in a different base. Wait, if it was base 2, then 2^8=256, but the coefficients are -1,0,1. But the problem is base 3. Hmm.Wait, perhaps the question has a mistake in the exponent range. If it was from i=1 to 8 instead of i=0 to 7, the maximum sum would be (3^9 -3)/2 = (19683 -3)/2=19680/2=9840, so non-negative numbers would be 9841. Still not matching.Alternatively, maybe the problem counts numbers that can be represented as positive numbers, excluding zero. That would be 3280 numbers. Still not matching.Wait, let's check the options again. (A)512=2^9, (B)729=3^6, (C)1094= maybe (3^7 +1)/2=1093 +1=1094, (D)3381, (E)59048.Ah! Wait, for n=7, the number of non-negative integers would be (3^7 +1)/2= (2187 +1)/2=1094. So that's option C. But in the problem, n=8. So if the problem had i=0 to 6 (7 terms), the answer would be C. But the problem says i=0 to7, which is 8 terms. So if there's a typo in the problem, and it was i=0 to6, then answer is C. Otherwise, perhaps the answer is D)3381, which might be (3^8 + something)/2. But 3^8 is 6561. (6561 + 201)/2= 3381. So if there was an addition of 201, but I don't see why.Alternatively, maybe the problem is considering a different kind of representation where coefficients can be -1,0,1, but with a different digit set. Wait, but the digit set is clearly given.Alternatively, maybe I need to consider that the digits are from 0 to7, but no, the digits are a_i.Wait, unless the problem is not in base 3, but each term is 3^i multiplied by a_i, but interpreted in base 10. Wait, that would be different. Wait, no, the problem says "represented as sum_{i=0}^7 3^i a_i", so it's a sum in base 10. Each term is 3^i * a_i, and a_i is -1,0, or1. So the total is just an integer calculated in base 10. So that is balanced ternary. So numbers from -3280 to 3280, 6561 numbers, with 3281 non-negative.Hmm. Maybe the answer is D)3381 due to a miscalculation by the problem's author. Alternatively, perhaps the original problem was in base 5 or something else. Alternatively, maybe the problem is in base 3, but digits can be 0,1,2, and the question is about something else. Wait, but the problem states a_i ∈ {-1,0,1}.Given that, and given that none of the options is 3281, perhaps the intended answer is D)3381, possibly due to a miscalculation where they thought the maximum value is 3^8 -1=6560, leading to non-negative numbers from 0 to6560, which would be 6561 numbers. But that would be if a_i ∈ {0,1,2}, regular ternary. But the problem says a_i ∈ {-1,0,1}. So if someone incorrectly assumes it's regular ternary with digits 0,1,2, then total numbers are 3^8=6561, from 0 to6560, so 6561 numbers. But that's not the case here. But option E is59048 which is much larger, and option D is3381.Wait, 3^8=6561. If you have digits -1,0,1, the total numbers are from -3280 to3280. If you have digits 0,1,2, it's from0 to6560. But the problem says digits are -1,0,1. So maybe the author of the problem made a mistake and thought it's regular ternary? If so, then the number of non-negative integers would be 3^8=6561, but that's option E)59048 is much larger. Wait, 3^8 is6561, which is option B)729 is 3^6.Alternatively, maybe there's a misunderstanding in the problem. If the question is about the number of non-negative coefficients (a_i), but that's not the case. The question is about the number of non-negative integers represented by the sum.Alternatively, perhaps the problem is asking for the number of solutions where the sum is non-negative, rather than the number of distinct sums. In that case, since each a_i can be -1,0,1, the number of sequences where the sum is non-negative would be (3^8 +1)/2=3281. But again, this is the same as the number of non-negative integers. So unless there's a different counting.Wait, but if different sequences can result in the same number, then the number of sequences where the sum is non-negative is 3^8 /2 + something. But since the numbers are symmetrically distributed around zero, except for zero itself, the number of sequences resulting in non-negative sums would be (3^8 +1)/2=3281. So that's again the same answer. So 3281.But since 3281 is not an option, perhaps the correct answer is D)3381 due to a miscalculation where someone thought the range is from 0 to 3^8 -1=6560, so non-negative integers are6561, but that's regular ternary. But the options don't have 6561 either. Option E is59048 which is way larger. Option D is3381. Maybe the problem is mixing up different bases or different digit sets.Alternatively, maybe the problem is considering base 3 digits 0,1,2, but transforming them into -1,0,1 by subtracting 1 from each digit. For example, if you have a digit 2, subtract 1 to make it 1, and carry over 1 to the next higher digit. This is similar to balanced ternary conversion. But if that's the case, then numbers can be represented in such a way, but the count would still be the same as balanced ternary. Hmm.Alternatively, maybe the problem is in base 9? Because 3^2=9. Wait, but the expression is sum 3^i a_i. So it's base 3.Given that all my reasoning leads to 3281, which is not an option, and given that the closest option is D)3381, I might have to consider that there's a mistake in the problem or the options. But since this is a multiple-choice question, and assuming that the options are correct, perhaps my approach is wrong.Wait, another approach: let's model this as a vector in a lattice. Each coefficient a_i contributes a vector of 3^i * a_i. The question is the number of non-negative points in this lattice. But this is similar to the number of non-negative integers representable by the sum.Alternatively, think of it as a walk where each step you can add -3^i, 0, or +3^i. The number of end positions that are non-negative.But how does this help? It might be complex.Alternatively, consider dynamic programming. Let's compute the number of distinct sums step by step.Start with sum = 0.For each power of 3 from 3^0 to 3^7, at each step, for each possible current sum, we can add -3^i, 0, or +3^i. So we need to track all possible sums.This is similar to a BFS approach where each step branches into three possibilities, but merging the same sums.The problem is that tracking all possible sums is computationally intensive, but perhaps we can find a pattern or use symmetry.At step 0 (i=0), possible sums are -1, 0, 1. Non-negative sums: 0,1 → count=2.At step 1 (i=1), for each previous sum s, we can add -3, 0, or +3. So for each s in {-1,0,1}, new sums are s-3, s, s+3.So possible sums after i=1:From s=-1: -4, -1, 2From s=0: -3, 0, 3From s=1: -2, 1, 4Total sums: -4, -3, -2, -1, 0, 1, 2, 3, 4. Non-negative sums: 0,1,2,3,4 → count=5.Similarly, at step 2 (i=2), we add -9, 0, +9 to each current sum. Current sums range from -4 to 4. After adding -9, ranges would be -13 to -5; adding 0 remains -4 to 4; adding +9 gives 5 to 13. Total sums from -13 to 13. Non-negative sums from 0 to13. The count would be previous non-negative count *3, but considering overlaps. Wait, no, overlaps can occur.But actually, each step adds a new layer that triples the previous number of sums, but with overlaps. However, due to the properties of balanced ternary, each number is uniquely representable, so the count should be 3^(i+1). Wait, no, for i=0, count=2; i=1, count=5; i=2, count=14? Wait, let's compute it step by step.Wait, for i=0:Sums: -1, 0, 1. Non-negative: 0,1. Count=2.i=1:From each sum in {-1,0,1}, add -3,0,3:-1-3=-4; -1+0=-1; -1+3=20-3=-3; 0+0=0; 0+3=31-3=-2;1+0=1;1+3=4Total sums: -4,-3,-2,-1,0,1,2,3,4. Non-negative: 0,1,2,3,4. Count=5.i=2:Add -9,0,9 to each previous sum from -4 to4.For each sum s in previous step:s-9, s, s+9.So new sums range from -4-9=-13 to 4+9=13.But need to check which of these are non-negative.For each sum in previous non-negative (0,1,2,3,4):Adding -9: 0-9=-9 (negative), 1-9=-8, ..., 4-9=-5 (all negative)Adding 0: 0,1,2,3,4Adding +9:9,10,11,12,13So new non-negative sums are:0,1,2,3,4 from adding 0; and 9,10,11,12,13 from adding +9. So total non-negative sums: 0-4 and 9-13. That's 5 +5=10 numbers. But wait, but balanced ternary up to i=2 should have (3^3 +1)/2=14 non-negative numbers. Wait, this approach is not matching.Wait, this suggests that dynamic programming approach might not account for all possible sums, but balanced ternary uniqueness ensures that all numbers in the range can be achieved. Therefore, maybe the dynamic programming approach is missing something.Alternatively, the discrepancy arises because when we add higher powers of 3, we can fill in the gaps between the existing numbers. For example, when we add 9 to the previous sums, we can reach numbers like 9-4=5, 9-3=6, etc. Wait, no, in the previous step, we were adding 9 to the existing sums. For example, if previous sum was 0, adding 9 gives 9; if previous sum was 1, adding 9 gives10, etc. But to get numbers like5,6,7,8, you need to combine lower powers.Ah, this is where the balanced ternary comes into play. By allowing negative coefficients for lower powers, you can reach numbers that are in between. For example, to get 5, you can have 9 - 3 -1 =5. So in i=2, you can represent 5 as +9 -3 -1. But in the dynamic programming approach, if you only consider adding -9,0,+9 to previous sums, you miss these combinations. Therefore, the dynamic programming approach is flawed because it doesn't allow modifying lower digits once they've been set.Therefore, the initial assumption that each number is uniquely representable in balanced ternary is correct, and thus the number of non-negative integers is indeed (3^8 +1)/2=3281. Since this is not an option, perhaps the problem is erroneous or there is a trick I'm missing.Wait, another thought: maybe the problem allows a_i to be -1,0,1, but without the restriction that the digits must form a valid balanced ternary number (i.e., allowing leading negative digits which might actually be positive when combined with higher digits). Wait, but in the standard balanced ternary, digits are unique. If you allow arbitrary combinations, then numbers can be represented in multiple ways, potentially increasing the number of non-negative integers. But no, even with multiple representations, the set of numbers that can be represented would still be from -3280 to 3280. Unless allowing higher digits to cancel out lower digits in a way that extends the range, which is not possible here.Wait, for example, if you have a higher digit set to +1 and lower digits set to -1's, you can get numbers beyond 3280. Wait, let's test that. Suppose a_7=1 and a_6 to a_0 are all -1. The sum would be 3^7 - (3^6 +3^5 +...+3^0). Let's compute:3^7 =2187Sum from i=0 to6 of3^i= (3^7 -1)/2=1093Therefore, total sum=2187 -1093=1094. Which is still within the -3280 to3280 range. So you can't exceed the maximum or minimum by combining digits. Therefore, the range is fixed.Thus, the number of non-negative integers remains 3281. Since this is not an option, and considering that option C is1094 which is (3^7 +1)/2, maybe the problem was supposed to be up to i=6 instead of i=7. In that case, the answer would be C)1094. But the problem states up to i=7. Without more information, I might have to assume that there's a mistake in the problem options, but since I have to choose from the given options, and knowing that the correct answer should be 3281, which is closest to D)3381, but I can't be sure. Alternatively, maybe I made a mistake in considering the range.Wait, let me recompute (3^8 +1)/2. 3^8=6561. 6561 +1=6562. 6562/2=3281. Yes. So that's correct.Given that, perhaps the answer intended is D)3381, but I can't see how to reach that number. Alternatively, maybe the question is asking for something else, like the number of non-negative coefficients across all representations, but that doesn't make sense.Wait, let's try another angle. If each a_i can be -1,0,1, then the total number of combinations is 3^8=6561. Each combination corresponds to a number between -3280 and3280. Now, how many of these 6561 numbers are non-negative? It's half of 6561 rounded up, which is (6561 +1)/2=3281. So that's the answer. Therefore, the problem's options are incorrect, or there's a mistranslation/misprint. Given that, and considering the closest option is D)3381, which is 100 more, maybe in the original problem, the exponent range was different, or there was a carryover error.Alternatively, if the problem had i from0 to 7 inclusive, that's 8 digits. If someone incorrectly used n=8 in the formula (3^(n+1) +1)/2, which would be (3^9 +1)/2=19684/2=9842, but that's not an option. Alternatively, maybe the problem is using base 2 instead of base3? No, because it's 3^i.Alternatively, if the problem had coefficients a_i ∈{0,1,2}, then the number of non-negative integers would be3^8=6561, which is option B)729=3^6, so no.Alternatively, if the problem is in base 4, but it's stated as3^i a_i, which is base3.Given that I can't reconcile the correct answer of3281 with the given options, and assuming that the problem expects the answer to be D)3381, I might have to consider that perhaps there's a different interpretation.Wait, one last try: perhaps the problem is asking for the number of non-negative coefficients sequences, i.e., the number of sequences where all a_i are non-negative. But since a_i can be -1,0,1, "non-negative integers" here refers to the sum being non-negative, not the coefficients. So that's not the case.Alternatively, maybe the problem is asking for the number of non-negative a_i's across all representations. But that also doesn't make sense.No, I think the problem is correctly stated, and my reasoning is correct, leading to3281, which is not an option. Therefore, there must be a mistake in the problem's options. However, since I have to choose from the given options, and considering that option C)1094 is (3^7 +1)/2, which would be the answer if the sum was up to i=6, maybe there was a misprint in the exponent range. If that's the case, then answer is C. However, the problem specifies up to i=7. Alternatively, the problem might have a different base.Given the provided options and the reasoning, I might have to go with option D)3381, even though it doesn't align with the calculations, assuming that perhaps there was a miscalculation or a different interpretation that I'm missing. But based on balanced ternary, the correct answer should be 3281, which is not listed. Therefore, I'm conflicted.Wait, let's check if 3381 is (3^8 + 3^7)/2. 3^8=6561, 3^7=2187. (6561+2187)/2=8748/2=4374. Not 3381.Alternatively, 3^8 - 3^7=6561-2187=4374. Not helpful.Alternatively, 3^8 - 3^4=6561-81=6480. No.Alternatively, 3^8 - 3^5=6561-243=6318.Not helpful.Alternatively, maybe the problem is counting numbers from 0 to3^8-1=6560, which would be6561 numbers. But that's regular ternary. If the problem had a_i ∈{0,1,2}, then answer would be6561. But that's not an option either. The closest is E)59048, which is way bigger.Alternatively, if the problem is in base 10, but it's using digits -1,0,1 multiplied by powers of3. No, that's not logical.Alternatively, the problem might be considering numbers where all digits are non-negative, i.e., a_i ∈{0,1}, which would be2^8=256 numbers. But option A is512=2^9. Not matching.Alternatively, maybe a_i can be 0,1, but with a carryover operation, making it base2. But no.Given all this, I think there's a mistake in the problem or the options. However, since the closest option to the correct answer (3281) is D)3381, and in some cases, question creators might have off-by-one errors or miscalculations, I might have to select D)3381. But I'm not confident. Alternatively, if the problem had i from0 to8, answer would be(3^9 +1)/2=9842, which is not an option. If i from0 to5, answer is (729 +1)/2=365, which is not an option. If i=0 to7, answer is3281. Since none of the options fit, but option C)1094 is(3^7 +1)/2, which would be the answer if the sum was up to i=6. Maybe the original problem had a typo. Given that, and if I had to choose the most plausible answer, it's C)1094, but the problem states up to i=7. This is very confusing.But since the given options don't include the correct answer, and the problem might be erroneous, I think the intended answer is D)3381. However, I cannot justify it mathematically. Alternatively, the problem might have a different interpretation that I haven't considered.Final Answerboxed{D}</think>The problem asks for the number of non-negative integers that can be represented as (sum_{i=0}^{7} 3^{i} a_{i}) where (a_{i} in {-1, 0, 1}). 1. Understanding the Representation: - Each term (3^i a_i) can be (-3^i), (0), or (3^i). - This is similar to balanced ternary representation, where each digit can be -1, 0, or 1.2. Range of Values: - The maximum value occurs when all (a_i = 1), which is (sum_{i=0}^{7} 3^i = frac{3^8 - 1}{2} = 3280). - The minimum value occurs when all (a_i = -1), which is (-3280).3. Number of Non-negative Integers: - The total number of distinct values from -3280 to 3280 is (3^8 = 6561). - Non-negative integers range from 0 to 3280, which is (frac{6561 + 1}{2} = 3281).4. Given Options: - The calculated value 3281 is not among the options provided (A) 512, (B) 729, (C) 1094, (D) 3381, (E) 59048.5. Re-evaluation: - Considering the closest option and potential errors in the problem statement or options, the closest plausible answer is (D) 3381, despite the discrepancy in calculation.boxed{D},<think>Okay, so I have this problem here: I need to find the number of non-negative integers that can be represented as a sum from i=0 to 7 of 3^i times a_i, where each a_i can be -1, 0, or 1. The answer choices are given, and I need to pick the correct one. Let me try to think through this step by step.First, let me parse the question. The expression is a sum over i from 0 to 7 of 3^i * a_i, and each a_i is in {-1, 0, 1}. So each term in the sum is either -3^i, 0, or 3^i. Then, the question is how many non-negative integers can be represented this way. The answer choices are 512, 729, 1094, 3381, and 59048.Hmm. So I need to figure out how many different non-negative numbers you can make by adding up these terms where each term is a power of 3 multiplied by -1, 0, or 1. Since each coefficient a_i can be -1, 0, or 1, this is similar to a number representation system. In fact, base 3 (ternary) representations use digits 0, 1, 2, but here we have digits -1, 0, 1. That might be called balanced ternary.Wait, balanced ternary! Yes, balanced ternary uses digits -1, 0, 1, often represented as digits -, 0, + or something like that. So maybe this problem is related to balanced ternary numbers. In balanced ternary, each digit position represents a coefficient for a power of 3, just like in this problem. So, perhaps the numbers representable here are exactly the numbers that can be written in 8-digit balanced ternary (from 3^0 up to 3^7). So the total number of such numbers would be... but wait, in balanced ternary, each number has a unique representation, right? Unlike standard ternary where you might have multiple representations with digits 0,1,2, balanced ternary is supposed to have unique representations for each integer within a certain range.But the question is asking for the number of non-negative integers that can be represented. So first, what is the range of numbers that can be represented with 8 digits in balanced ternary? Let's see. The maximum number would be when all a_i are 1: sum_{i=0}^7 3^i *1 = (3^8 -1)/2. Because the sum of 3^0 + 3^1 + ... +3^7 is a geometric series. The sum is (3^8 -1)/2. Let me check that formula. The sum S = 1 + 3 + 3^2 + ... +3^n is (3^{n+1} -1)/2. So here, n=7, so S = (3^8 -1)/2 = (6561 -1)/2 = 6560/2 = 3280. Wait, 3^8 is 6561, right? 3^1=3, 3^2=9, 3^3=27, 3^4=81, 3^5=243, 3^6=729, 3^7=2187, 3^8=6561. So yes, sum from i=0 to 7 is 1 + 3 + 9 + 27 + 81 + 243 + 729 + 2187 = Let's compute that step by step: 1+3=4, 4+9=13, 13+27=40, 40+81=121, 121+243=364, 364+729=1093, 1093+2187=3280. Yep, that matches (3^8 -1)/2 = 6560/2 = 3280. So the maximum positive number is 3280, and the minimum number (if all a_i are -1) is -3280. But since the question is about non-negative integers, we need to count how many integers from 0 up to 3280 can be represented.But balanced ternary can represent every integer in that range, right? Because balanced ternary allows unique representations for each integer from -N to N, where N is (3^k -1)/2 for k digits. Wait, in this case, with 8 digits, the range should be from -3280 to 3280, and each integer in that range has a unique balanced ternary representation. Therefore, the total number of integers representable is 3280 - (-3280) +1 = 6561. But 6561 is 3^8, which makes sense because each digit has 3 choices, so 3^8 total representations. But in standard balanced ternary, you have unique representations for each integer in that range, so 6561 numbers, from -3280 to 3280 inclusive. But the question is only about non-negative integers, which would be from 0 to 3280 inclusive. So how many numbers is that? 3280 - 0 +1 = 3281. But 3281 isn't one of the answer choices. The options are 512, 729, 1094, 3381, 59048.Hmm. 3281 is close to 3381, which is option D. So perhaps I made a mistake here. Let me check again.Wait, maybe my assumption that balanced ternary represents each integer in that range uniquely is correct, but maybe the question is not restricting to unique representations? Wait, the problem says "can be represented as", so even if a number can be represented in multiple ways, we just need to count it once. But in balanced ternary, each number has exactly one representation, so the total number of numbers from -3280 to 3280 is 3^8 = 6561. Therefore, non-negative numbers would be 3280 +1 = 3281. But 3281 is not an option here. The closest is 3381. Hmm. So either my reasoning is wrong, or perhaps the question is not about balanced ternary?Wait, let me check the problem again. The problem says "non-negative integers that can be represented as sum_{i=0}^7 3^i a_i where a_i ∈ {-1,0,1}". So each a_i is -1, 0, or 1, but the sum is non-negative. So even if in balanced ternary, numbers are uniquely represented, but here maybe coefficients can be chosen in different ways, even if they are not the balanced ternary digits. Wait, no: balanced ternary uses digits -1, 0, 1, so this is exactly balanced ternary. So each number from -3280 to 3280 can be uniquely represented as such a sum, so there's 6561 numbers total, half negative, half positive, and zero. Wait, but 3280*2 +1=6561. Therefore, non-negative integers are from 0 to 3280 inclusive, which is 3281 numbers. But 3281 is not an option. The options are (A) 512, (B)729, (C)1094, (D)3381, (E)59048.Hmm. So 3281 is not there, but 3381 is. Maybe my calculation is wrong? Wait, let's compute 3^0 to 3^7 again. 3^0=1, 3^1=3, 3^2=9, 3^3=27, 3^4=81, 3^5=243, 3^6=729, 3^7=2187. Sum these up: 1+3=4, +9=13, +27=40, +81=121, +243=364, +729=1093, +2187=3280. So the total is 3280. So maximum number is 3280, so numbers from 0 to 3280 inclusive. That's 3281 numbers. But answer choice D is 3381, which is 100 more. So maybe there's a mistake here.Wait, maybe the problem is not using balanced ternary? Wait, but the coefficients are -1, 0, 1. That's exactly balanced ternary. But perhaps the question is not about unique representations? Wait, no. If you allow coefficients -1,0,1, then each number can be represented in balanced ternary uniquely. But maybe the problem allows leading zeros? Wait, but even with leading zeros, the number of representations would still be unique if you consider the digits from 0 to 7. Wait, but in standard positional numbering systems, leading zeros don't change the value. So for example, 00000001 in balanced ternary is still 1. So even with leading zeros, the value remains the same. Therefore, the number of distinct numbers is still 3^8, but they map to integers from -3280 to 3280. So non-negative integers would be 0 to 3280, which is 3281 numbers. Since 3281 is not an option, perhaps the problem is not about balanced ternary?Wait, another approach. Maybe this is similar to binary representations but with base 3 and digits -1,0,1. So the total number of possible combinations is 3^8=6561. Each combination corresponds to a number. However, some combinations might result in the same number, so the total number of distinct numbers is less than 6561. But balanced ternary is a unique representation system, so each number from -3280 to 3280 is represented exactly once. Therefore, the number of non-negative integers would be 3281. But since 3281 is not an answer choice, perhaps the question is interpreted differently.Wait, maybe the problem is allowing a_i to be -1,0,1 for each i from 0 to 7, but the total sum is non-negative. So we need to count how many different sums (allowing different combinations of a_i) result in a non-negative integer. However, since each combination of a_i's gives a unique number (due to balanced ternary), the number of non-negative integers would be exactly half of 3^8 rounded up. Because for every positive number, there's a corresponding negative number, and zero is in the middle. 3^8 is 6561. Half of that is 3280.5, so 3281 non-negative numbers. But again, 3281 is not an option. Wait, but the answer choices don't have 3281. The closest is 3381. Maybe the question is considering something else.Wait, perhaps the problem is not using signed digits but absolute values? Wait, no, the coefficients can be -1,0,1, so the digits are signed. Wait, maybe the question is asking for the number of non-negative integers representable with digits a_i in {-1,0,1}, but allowing multiple representations? But no, even if you allow multiple representations, the number of distinct integers would still be the same as balanced ternary, since balanced ternary already covers all those numbers uniquely.Alternatively, perhaps the problem is not using balanced ternary but something else. Let me think. Suppose each digit can be -1,0,1, but when you write the number, you have to use exactly 8 digits (i.e., including leading zeros). But in that case, numbers would still be from -3280 to 3280. Wait, but how does leading zeros affect it? If you have leading zeros, they don't contribute to the value. So even if you have a_i=0 for higher i's, the value is the same as not having those digits. Wait, but in the problem statement, the sum is from i=0 to 7, so each term is included, even if a_i=0. So for example, even if all a_i=0 except a_0=1, it's 1. If all a_i=0 except a_7=1, it's 3^7=2187. So the total number is 3^8=6561 different sequences of a_i's, each corresponding to a number. However, some sequences might produce the same number. Wait, but in balanced ternary, each number has a unique representation. So even if you have leading zeros, the value is still unique. Therefore, the total numbers representable are 3^8=6561, but since each sequence corresponds to a unique number (with possible leading zeros), those numbers span from -3280 to 3280. Therefore, non-negative numbers are 3281. But since 3281 is not an option, maybe there is a miscalculation?Wait, let's compute 3^8: 3^8=6561. So the total number of different sequences is 6561. Each of these sequences corresponds to a unique number in balanced ternary. So numbers from -3280 to 3280 inclusive. Therefore, non-negative integers are 0 to 3280, which is 3281 numbers. But the answer choices are 512, 729, 1094, 3381, 59048. Hmm. 3381 is 3280 + 101. Wait, perhaps there's a misunderstanding in the problem. Maybe the problem is not about balanced ternary, but about something else. Let me read the problem again."18. There are ( ) non-negative integers that can be represented as sum_{i=0}^7 3^i a_i (a_i ∈ {-1,0,1}, 0 <=i <=7)."So the question is: how many non-negative integers can be expressed in the form of that sum where each coefficient a_i is -1, 0, or 1. So each a_i is independent. So each term is either subtracting 3^i, leaving it out, or adding it. So the total number of possible combinations is 3^8=6561, but many of these combinations result in the same integer. Wait, but earlier I thought balanced ternary gives unique representations. But if that's the case, then the total number of integers is 6561, but that's impossible because the range is from -3280 to 3280, which is 6561 numbers. Wait, but 3280 - (-3280) +1 = 6561. So that's exactly the number of integers in that interval. Therefore, each combination corresponds to a unique integer. Therefore, the number of non-negative integers is (6561 +1)/2 = 3281, since from -3280 to 3280, inclusive, there are 6561 numbers, so half are negative, half are positive, and zero. So non-negative would be 3281. But 3281 is not an option. Hmm.Wait, but maybe the problem is not counting integers, but non-negative numbers, including all non-negative integers that can be represented. Wait, but if the representation is unique, then the count should be 3281. However, perhaps the problem is considering numbers that can be represented in any way, not necessarily uniquely. Wait, but even if you consider multiple representations, the total number of distinct numbers is still 6561. So maybe the answer is 3281, but since that's not an option, maybe the question is differently intended.Wait, let me check the answer choices again: 512 is 2^9, but 3^8 is 6561. 729 is 3^6, which is 729. 1094 is maybe 3^7 + something? 3^7 is 2187. Hmm. 3381 is close to 3^8 / 2, which is 3280.5, approximately 3281. But 3381 is 100 more. 59048 is 3^10 - something? Wait, 3^10=59049, so 59048 is one less. Not sure.Alternatively, maybe the problem is not considering the balanced ternary's unique representation but is instead allowing coefficients to be -1, 0, 1 in each digit, but numbers can be represented in different ways, hence overlapping. Wait, but in standard ternary (digits 0,1,2), numbers can be represented in only one way, but in balanced ternary, also unique. Wait, maybe there is a different interpretation here.Alternatively, maybe the problem is not about integers but about the number of non-negative values, including duplicates. But the question says "non-negative integers", so duplicates shouldn't matter; it's the count of distinct integers. So perhaps 3281 is the answer, but it's not listed. Then, perhaps I made a mistake in calculating the maximum value.Wait, the maximum value is sum_{i=0}^7 3^i *1. Let's compute that again. 3^0 + 3^1 + ... + 3^7. Let's compute step by step:3^0 =11 +3=44 +9=1313 +27=4040 +81=121121 +243=364364 +729=10931093 +2187=3280Yes, 3280. So the maximum is 3280, so the numbers go from -3280 to 3280, inclusive. That's 6561 numbers. Therefore, non-negative is 3281. But the answer choices are A)512, B)729, C)1094, D)3381, E)59048. None of these is 3281. So something is wrong here.Wait, maybe the problem is not about 3^i for i from 0 to 7, but 3^i for i from 0 to 7, which is 8 digits. Wait, but 3^7 is the highest term. So the maximum value is 3280, as before. Hmm.Wait, maybe the problem is not considering negative coefficients? Wait, no, the problem allows a_i to be -1,0,1, so negative coefficients are allowed, but the sum has to be non-negative. So we need to count the number of different sums (each sum being a non-negative integer) that can be formed by choosing each a_i as -1,0,1. But the key is that different choices of a_i's can lead to the same sum. So the total number of non-negative integers is not simply 3^8 /2 rounded up, but rather the number of distinct sums that are non-negative.But in balanced ternary, each number from -3280 to 3280 has a unique representation. Therefore, each a_i sequence corresponds to a unique number. Therefore, the number of non-negative integers is exactly the number of sequences where the sum is non-negative. Since each sequence is unique, half of them (rounded up) would be non-negative. So total sequences: 3^8=6561. Then the number of non-negative integers would be (6561 +1)/2=3281. Because from -3280 to 3280 inclusive, there are 6561 numbers. The number of non-negative numbers (including zero) is 3280 +1=3281. So 3281. But again, 3281 is not an option. So perhaps the question is not about balanced ternary, but about a different system where digits can be -1,0,1 but with some constraints?Wait, maybe the problem is similar to binary, where each digit can contribute -1,0,1, but the numbers are being counted as absolute values. Wait, no. Let me think differently.Alternatively, maybe the question is asking for the number of non-negative integers without considering uniqueness. But that doesn't make sense, because the problem says "there are ( ) non-negative integers that can be represented", so it's about the count of integers, not the count of representations.Wait, maybe the problem is in Chinese, and there was a translation error? The original problem number 18. Maybe the exponents are from 1 to 8 instead of 0 to 7? Let's check. If the exponents were from 1 to 8, then the sum would be 3^1 a_1 + ... + 3^8 a_8. Then the maximum sum would be 3^1 + ... +3^8 = (3^9 -3)/2. Let's compute that: 3^9=19683, so (19683 -3)/2=19680/2=9840. Then the range would be from -9840 to 9840, so non-negative numbers would be 9841, which is still not among the options. Hmm.Alternatively, maybe the question is not about 3^i but about (-3)^i? But that would complicate things, but the problem says 3^i a_i, so each term is 3^i multiplied by a coefficient -1,0,1.Wait, maybe the problem is considering variable length representations? For example, allowing leading zeros beyond i=7? But no, the problem states the sum is from i=0 to 7, so it's fixed at 8 digits.Wait, perhaps the problem is in base 3, but digits can be -1,0,1, and leading zeros are not allowed? But leading zeros don't affect the value. Even if leading zeros are disallowed, but in this problem, each term is included, so even if a_7=0, you still have that term as 0*3^7. So it's like fixed 8 digits. So leading zeros are allowed. So each number is uniquely represented with 8 digits, so the total numbers are 3^8=6561, from -3280 to 3280.So, given that, why isn't 3281 an answer option? Maybe the problem is different. Let me check the original problem again.Wait, the problem is presented as a multiple-choice question with options (A)512, (B)729, (C)1094, (D)3381, (E)59048. The problem is in Chinese, but the translation is accurate? The user provided the translation, so assuming it's accurate. The problem is asking for the number of non-negative integers representable as the sum from i=0 to 7 of 3^i a_i where a_i ∈ {-1,0,1}.Wait, perhaps the answer is 3^8 = 6561, but since we need non-negative integers, it's half of that, rounded up. 6561 /2 = 3280.5, so 3281. But 3281 is not listed, but 3381 is. Hmm. Alternatively, maybe the problem is not considering that each number can be represented uniquely, but instead counts the number of non-negative numbers that can be expressed, possibly with different representations. But even so, the total number of distinct numbers would still be 6561, with 3281 non-negative. So this still doesn't resolve the discrepancy.Wait, maybe there's a miscalculation in the maximum number. Wait, let me compute the maximum again. If each a_i is 1, the sum is 3^0 + 3^1 + ... +3^7. Let's compute this:3^0 =13^1=3; cumulative sum:1+3=43^2=9; cumulative sum:4+9=133^3=27; cumulative sum:13+27=403^4=81; cumulative sum:40+81=1213^5=243; cumulative sum:121+243=3643^6=729; cumulative sum:364+729=10933^7=2187; cumulative sum:1093+2187=3280. Yes, that's correct. So the maximum is indeed 3280, so non-negative numbers are 0 to 3280 inclusive, which is 3281 numbers.Wait, perhaps the problem is in base 3 but using digits 0,1,2, but with the ability to subtract. Wait, no, the coefficients are -1,0,1. Wait, unless the problem is actually in base 3 but allowing digits to be -1,0,1, but with a different carryover system. But balanced ternary does use digits -1,0,1, with digits adjusted during addition/subtraction. However, in standard balanced ternary, each number has a unique representation. So unless there's a different interpretation here.Alternatively, maybe the problem is about the number of non-negative coefficients in all possible sums, but that doesn't make sense. Wait, no. The problem is asking for the number of non-negative integers that can be represented as such a sum. Each sum is an integer, and we need to count how many distinct integers are non-negative.Given that balanced ternary with 8 digits can represent all integers from -3280 to 3280, inclusive, that's 6561 numbers. Thus, the number of non-negative integers is 3281. But the answer isn't there. Hmm.Wait, maybe the problem is not using balanced ternary but something else. Let me think differently. Suppose we think of each coefficient a_i as a digit in base 3, but shifted by -1. So instead of digits 0,1,2, we have digits -1,0,1. In standard base 3, the value is sum_{i=0}^7 3^i * d_i where d_i ∈ {0,1,2}. The number of non-negative integers in standard base 3 with 8 digits is 3^8=6561, from 0 to 6560. But in this problem, the digits are shifted. So perhaps the range is shifted?Wait, if we let b_i = a_i +1, then each b_i ∈ {0,1,2}, so the sum becomes sum_{i=0}^7 3^i*(b_i -1) = sum_{i=0}^7 3^i*b_i - sum_{i=0}^7 3^i. The first sum is a standard base 3 number with digits 0,1,2, which ranges from 0 to 6560. The second sum is (3^8 -1)/2 = 3280. Therefore, the total sum is (standard base 3 number) - 3280. Therefore, the possible values of the original sum range from -3280 to 6560 - 3280 = 3280. Wait, that's interesting. So this would mean that by shifting the digits, the range becomes from -3280 to 3280, same as balanced ternary. Wait, but in that case, the number of distinct integers would still be 6561. Therefore, the number of non-negative integers would be 3281. But again, that's not an option. Hmm.Alternatively, maybe the problem is considering the total number of non-negative numbers, not just integers. But the problem specifically says "non-negative integers". So decimals are not considered. So the answer should be 3281. But since that's not an option, maybe the problem is intended to have a different solution.Wait, let's try another approach. Maybe we can model this as a generating function. The generating function for each digit a_i would be (x^{-3^i} + x^{0} + x^{3^i}). So the total generating function is the product from i=0 to 7 of (x^{-3^i} + 1 + x^{3^i}). Then, the coefficient of x^k in this product would be the number of ways to represent k as such a sum. Then, the problem is asking for the number of k >=0 for which the coefficient is at least 1. So, the question reduces to finding the number of non-negative integers k where the coefficient of x^k is non-zero.But calculating this generating function and finding the number of non-zero coefficients for k >=0 might be complex, but perhaps we can find a symmetry or use some properties.Note that the generating function is symmetric around x^0 because for each term (x^{-3^i} +1 +x^{3^i}), it's symmetric. So the entire generating function is symmetric. Therefore, the number of non-negative coefficients would be equal to the number of non-positive coefficients, and the zero coefficient is counted once. Therefore, the total number of non-negative integers would be (number of non-zero coefficients)/2 +1 (for zero). But how many non-zero coefficients are there?But the total number of terms in the generating function is 3^8=6561, but many coefficients overlap. However, in balanced ternary, each coefficient corresponds to a unique integer, so there's no overlap. Therefore, the number of non-zero coefficients is 6561 (since all sums are unique). Wait, but no. If we have 8 digits each contributing -3^i, 0, or 3^i, the sums can range from -3280 to 3280, which is 6561 distinct integers. Therefore, all coefficients in the generating function are either 0 or 1, and each integer in that range is represented exactly once. Therefore, the number of non-negative integers is 3281. But this again brings us back to the same number, which is not an option.Given that none of the answer choices match 3281, and the closest is 3381, perhaps there is an error in the problem statement or the answer choices. Alternatively, maybe my entire approach is wrong.Wait, another thought. Maybe the problem is not considering the uniqueness of representations. For example, in standard base 3 with digits 0,1,2, each number has a unique representation, but if we allow digits -1,0,1, then some numbers might have multiple representations. For example, maybe 1 can be represented as 1*3^0 or 1*3^1 -1*3^0 -1*3^0 or something. Wait, no. In balanced ternary, each number has a unique representation. For example, 1 is represented as 1*3^0. 2 is represented as 3^1 -1*3^0. 3 is 1*3^1. So each number has a unique representation, so the coefficients are uniquely determined. Therefore, even if you allow digits -1,0,1, each number is represented once. Therefore, the total number of numbers is 6561, with non-negative numbers being 3281. But this is not an answer choice. Therefore, maybe the problem is not in base 3?Wait, but the terms are explicitly 3^i multiplied by a_i. So that is base 3. Wait, unless there is a different base involved. Wait, the problem is written as sum_{i=0}^7 3^i a_i, so base 3 digits, with each digit being -1,0,1. Hence balanced ternary. So, unless the question is in error or the answer choices are wrong, there's a disconnect here.Alternatively, maybe the problem is not about the number of distinct integers but the number of non-negative values, counting different representations. But that would just be 3^8=6561, but the non-negative ones would be half, roughly 3280.5, so 3281. Still not matching the options.Wait, let's look at the answer choices again: (A)512=2^9, (B)729=3^6, (C)1094, (D)3381, (E)59048=3^10-1.Hmm. 729 is 3^6. If the problem was from i=0 to 5 (6 digits), then the total numbers would be 3^6=729, with non-negative numbers being (729 +1)/2=365, which is not 729. Not sure.Alternatively, if the problem had coefficients in {0,1,2}, then the number of non-negative integers would be 3^8=6561, but that's not the case here.Alternatively, if the digits were from 0 to 7 instead of i=0 to7, but no, the problem states i=0 to7.Alternatively, maybe the problem is a trick question. Since each a_i can be -1,0,1, the total number of combinations is 3^8=6561. However, some combinations produce the same integer. But since balanced ternary is a unique representation system, each combination corresponds to a unique integer. Therefore, the number of non-negative integers is (6561 +1)/2=3281. But since this isn't an option, perhaps the problem is a different type.Wait, another angle. Perhaps the question is not about the range but about the number of numbers that can be expressed as such a sum where each digit is independently chosen as -1,0,1. However, in this case, the total number of possible sums is 3^8=6561, but each sum is an integer between -3280 and 3280. However, the number of distinct integers is exactly 6561, which is from -3280 to 3280 inclusive. Therefore, non-negative integers would be 3281. But since this is not an answer choice, perhaps there is a different interpretation.Wait, maybe the question is asking for the number of non-negative coefficients across all possible sums, but that seems not to make sense.Alternatively, maybe the question is in error, or there is a mistranslation.Alternatively, perhaps the problem is considering all possible subsets where you can add or subtract 3^i terms, and wants the number of non-negative integers. Wait, that's the same as balanced ternary. So it should still be 3281.Alternatively, maybe there's a different way to compute this. Let's consider dynamic programming. Let's define dp[i][j] as the number of ways to reach sum j using the first i digits. But since we need the number of non-negative integers, we need the count of j >=0 that can be formed.But calculating this for i=8 and j up to 3280 might be tedious, but perhaps we can find a pattern. However, since balanced ternary is a unique representation system, each number is formed exactly once, so the total number is 3^8=6561 numbers, with 3281 non-negative. Therefore, the answer should be 3281. Since this is not among the choices, perhaps the intended answer is 3^8 / 2 rounded up, but 6561 /2=3280.5, which is 3281, which is still not an option.Wait, but let's check the answer choices again. (C) 1094. 1094 is approximately 3^7=2187, so no. (D) 3381. If we compute 3^8 /2 + something. 3^8=6561. 6561 /2=3280.5. 3280.5 +100=3380.5≈3381. Maybe the problem has a typo, and instead of i=0 to7, it's i=0 to8. Let's check. If it's i=0 to8, then the maximum sum is (3^9 -1)/2=(19683 -1)/2=19682/2=9841. Then the number of non-negative integers would be 9841 +1=9842. Not an option either.Alternatively, maybe the problem is considering that each a_i can be -1,0,1, but not allowing the digit -1 in the highest position (i=7), to avoid negative numbers. But that complicates it. For example, if a_7 is restricted to 0 or1, then the maximum number is 3^7 + ... +3^0 =3280, and the minimum is - (3^7 -1)/2= - (2187-1)/2= -1093. Then the total numbers would be 3280 +1093 +1=4374. Then non-negative numbers would be from 0 to3280, which is3281, still not matching.Alternatively, if the problem is allowing variable length representations, not fixed to 8 digits. But the problem states sum_{i=0}^7, so fixed length.Alternatively, maybe the problem is not about integers but about natural numbers, excluding zero. But 3280 would still not be an option.Alternatively, maybe the problem is in base 9, but the coefficients are -1,0,1. But the terms are 3^i, not 9^i.Wait, another idea: perhaps the problem is similar to a base-3 expansion where digits can be -1, 0, or 1, but with the caveat that numbers can be represented in multiple ways, and the count is over the number of such numbers. However, in balanced ternary, each number has a unique representation. So this still results in 3281.Alternatively, perhaps the problem is thinking in terms of the number of solutions to the equation sum_{i=0}^7 3^i a_i >=0, where a_i ∈ {-1,0,1}. But that's the same as the number of sequences (a_0,...,a_7) such that the sum is non-negative. Since each sequence corresponds to a unique number, the number of non-negative numbers is equal to the number of sequences where the sum is non-negative, which is (3^8 +1)/2=3280.5, rounded up to 3281.But since none of the answer choices match, I'm at a loss. Given the answer choices, perhaps the intended answer is 3^6=729, but that doesn't make sense. Or 3^7=2187, but that's not an option either. The closest is 3381, which is D. Maybe there's a different approach.Wait, let's think about the problem as a lattice in an 8-dimensional space where each dimension can be -1,0,1. The question is how many points in this lattice have a non-negative sum. The total number of points is 3^8=6561. If we assume that the distribution of sums is symmetric around zero, then the number of non-negative sums would be (6561 +1)/2=3281. But since the answer choices don't include this, maybe the symmetry assumption is incorrect?Wait, in balanced ternary, the representations are unique, so the distribution is symmetric. Therefore, the number of non-negative numbers is (6561 +1)/2=3281. Therefore, if this is not an option, maybe the question is flawed or there's a different interpretation.Alternatively, perhaps the problem is considering that some numbers can be represented in more than one way, hence leading to more numbers. But balanced ternary's uniqueness contradicts that. Unless the system is different.Wait, let's take a smaller case. Suppose we have i=0 to1 (2 digits). Then the possible sums are:-1*3^0 -1*3^1 = -1 -3 = -4-1*3^0 +0*3^1 = -1-1*3^0 +1*3^1 = -1 +3 = 20*3^0 -1*3^1 = -30*3^0 +0*3^1 =00*3^0 +1*3^1=31*3^0 -1*3^1=1 -3=-21*3^0 +0*3^1=11*3^0 +1*3^1=1+3=4So the possible sums are: -4, -3, -2, -1, 0,1,2,3,4. That's 9 numbers, which is 3^2=9. The non-negative numbers are 0,1,2,3,4: 5 numbers. Which is (9 +1)/2=5. So for n=2, it works. Similarly, for n=1, i=0: sums are -1,0,1. Non-negative: 0,1: 2 numbers, which is (3 +1)/2=2. So the formula holds. Therefore, for n=8, it should be (3^8 +1)/2=3281. So unless there's a specific reason here, this should be the case.Given that the answer choices do not include 3281, maybe the problem had a typo, and the correct answer is D)3381, but I can't see how. Alternatively, maybe the question allows digits beyond i=7, but it's stated as up to i=7. Alternatively, maybe the exponents start at i=1. Let's check that.If the sum is from i=1 to8, then the max sum is (3^9 -3)/2=(19683 -3)/2=19680/2=9840. Then non-negative numbers would be 9841. Not an option. If the sum is from i=0 to6, then max is (3^7 -1)/2=1093. Non-negative numbers would be 1094. Oh! Option C is 1094. Wait, but the problem says from i=0 to7. If it was i=0 to6, the answer would be C)1094. So perhaps there was a misprint in the problem, stating i=0 to7 instead of i=0 to6. But since the original problem is in Chinese, maybe there was a translation error or a misprint.Alternatively, maybe the problem is correct as stated, but I'm missing something. Wait, let's compute the number of non-negative integers for i=0 to6. Then the maximum sum is (3^7 -1)/2=1093, so non-negative numbers are 1094, which is option C. If the problem had said from i=0 to6, then answer C would be correct. But the problem states i=0 to7. So perhaps the answer should be 3281, but it's not listed. Therefore, there might be an error in the problem or the answer choices.Given that the closest option to 3281 is 3381 (D), which is 100 more, maybe there is a miscalculation. Alternatively, maybe the problem counts zero as a positive integer, but no, zero is non-negative.Alternatively, maybe the problem is not about balanced ternary but about something else. Let's think again. If we treat each a_i as a digit in a number system where digits can be -1,0,1, but without the balanced ternary rules (i.e., just allowing any combination of digits), then overlaps can occur. For example, the number 1 can be represented as 1*3^0, or as 1*3^1 -1*3^0 -1*3^0, etc. But in reality, in balanced ternary, digits are unique. However, if we don't follow the balanced ternary rules, maybe numbers can be represented in multiple ways, but even so, the set of numbers that can be represented would still be from -3280 to 3280. The number of distinct numbers would still be 6561, hence non-negative numbers would be 3281.Alternatively, maybe the problem is asking for the number of non-negative numbers representable with at most 8 digits, but allowing shorter lengths. However, even then, the maximum number is still 3280. So non-negative numbers would still be 3281.Alternatively, maybe the problem is in base 3, but using digits -1,0,1 for each position, but allowing the numbers to be of any length, not limited to 8 digits. But the problem states the sum is from i=0 to7, so it is limited.Given all this, I think there might be an error in the problem's answer choices or in the problem statement itself. However, given the closest option is D)3381, which is 100 more than my calculated 3281, maybe there's a different approach.Wait, let's consider that when you have digits from 0 to7, maybe the carryover in balanced ternary allows for more numbers. Wait, no. Balanced ternary is a positional numeral system where each digit must be -1,0,1, and numbers are uniquely represented. So even with multiple digits, each number is unique.Alternatively, maybe the problem is not restricting the number of digits and allowing any number of digits. But the problem says sum from i=0 to7, so it's restricted.Alternatively, maybe the problem is considering that a_7 can be only 0 or1 to ensure the number is non-negative. If we force a_7 to be 0 or1, then the total number of combinations is 2*3^7=4374. The number of non-negative integers would then be 4374, but that's not an option either.Alternatively, maybe the problem is considering a different base. If we think in base 3, but using digits -1,0,1, but for base conversion, but I don't see how that changes the count.Wait, another idea: maybe the problem is asking for the number of non-negative coefficients across all possible sums, but that interpretation doesn't fit the wording.Alternatively, maybe the question is in error and the correct answer is 3281, but it's not listed. In that case, maybe the intended answer is D)3381, and there's a mistake in the problem's exponent range or the answer choices.Given that this is a multiple-choice question, and given my calculation repeatedly gets 3281, which isn't an option, but the closest is D)3381, maybe there is a typo in the problem or the answer choices. Alternatively, perhaps the original problem was in Chinese and there was a mistranslation, such as the exponent range being different.Alternatively, perhaps the problem is not about balanced ternary but about another system where coefficients can be -1,0,1 but digits can be repeated in a way that allows more numbers. But without more information, it's hard to see.Given that in similar problems, sometimes the number of non-negative numbers in balanced ternary with n digits is (3^n +1)/2. For n=8, this would be (6561 +1)/2=3281. This formula is known in combinatorics for balanced ternary systems. Therefore, I believe the correct answer should be 3281. But since that's not an option, I'm inclined to think there's an error in the problem. However, given the answer choices, perhaps the intended answer is D)3381, possibly due to a miscalculation or a different interpretation.But since 3281 is not available, and given that 3^8=6561, which is the total number of representations, and maybe the problem is asking for the total number of non-negative representations, which would be half of 6561, approximately 3280.5, which is 3281, but since it's not present, maybe the answer intended is D)3381. But I cannot justify this with the given information.Alternatively, perhaps the problem counts zero as a positive integer, but even then, it would still be 3280 non-zero non-negative integers, plus zero, totaling 3281.Given the available options and the reasoning, I might have to consider that there's a mistake in the problem or answer choices. However, since the closest option is D)3381, maybe that's the intended answer. But I'm not sure. Alternatively, if I consider the maximum number as 3^8 instead of (3^8 -1)/2, then the maximum would be 3^8=6561, but that's not possible with coefficients of -1,0,1.Wait, if all a_i=1, sum= (3^8 -1)/2=3280. If all a_i=1 except one being -1, then sum=3280 -2*3^i. Wait, but that's still less than 3280. So no, the maximum is 3280.Alternatively, if the problem is in base 3 but using digits 0,1,2, then the number of non-negative integers would be 3^8=6561, but that's not the case here.Given all this, I think the correct answer based on balanced ternary should be 3281, which is not listed, but the closest is D)3381. However, since the answer choices are given, and assuming there's an error, but if I have to choose from the options, maybe the answer is D)3381. But I'm not confident.Alternatively, maybe the problem is not using balanced ternary but a different encoding where digits can be -1,0,1 but with a different weight. For example, maybe it's a different base. But the problem states 3^i, so it is base 3.Alternatively, maybe the problem is about the number of non-negative numbers without considering that some numbers can't be represented. For example, in base 3 with digits -1,0,1, maybe numbers congruent to certain residues modulo something can't be represented. But balanced ternary can represent all integers in the range, so every integer is representable.In conclusion, based on balanced ternary, the number of non-negative integers should be 3281, but since it's not an option, I might have to assume a typo and choose the closest option D)3381, but I can't confirm this. Alternatively, the problem might have a different intended solution that I haven't considered.Final Answerboxed{D}</think>The problem requires finding the number of non-negative integers that can be represented as (sum_{i=0}^{7} 3^{i} a_{i}) where (a_{i} in {-1, 0, 1}). 1. Balanced Ternary Representation: The given expression is equivalent to balanced ternary representation, where each digit (a_i) can be -1, 0, or 1. In balanced ternary, each integer within a specific range can be uniquely represented.2. Range Calculation: The maximum value for the sum is calculated by setting all (a_i = 1), resulting in (sum_{i=0}^{7} 3^i = frac{3^8 - 1}{2} = 3280). Similarly, the minimum value (all (a_i = -1)) is -3280. Thus, the range of representable integers is from -3280 to 3280.3. Count of Non-Negative Integers: The total number of integers in this range is (6561) (from -3280 to 3280 inclusive). The number of non-negative integers (including zero) is (frac{6561 + 1}{2} = 3281).4. Answer Choices: The given answer choices are 512, 729, 1094, 3381, and 59048. The calculated value 3281 is not listed, but the closest option is 3381 (D).Given the closest available answer and potential considerations for alternative interpretations or possible errors in the problem statement, the final answer is:(boxed{D})]

🔑:18. D.Since sum_{i=0}^{7} 3^{i} a_{i} is a non-negative integer, the highest bit cannot be negative.We discuss this by cases.a_{7}=1, there are 3^{7} cases;a_{7}=0, a_{6}=1, there are 3^{6} cases;a_{7}=0, a_{6}=0, a_{5}=1, there are 3^{5} cases;qquada_{7}=a_{6}=cdots=a_{1}=0, a_{0}=1, there is 1 case.Therefore, the desired result is sum_{i=0}^{7} 3^{i}=3281.

🧠:[<think>Okay, so I have this problem here that says: No matter what the real value of m is, the line y + 1 = m(x - 2) always passes through a fixed point. The coordinates of this point are one of the options given, which are A: (1, -2), B: (-1, 2), C: (-2, -1), D: (2, -1). Hmm, interesting. Let me try to figure this out step by step.First, the equation given is y + 1 = m(x - 2). This looks like the point-slope form of a line. In general, point-slope form is y - y1 = m(x - x1), where (x1, y1) is a point the line passes through and m is the slope. Wait, but in this case, the equation is y + 1 = m(x - 2). So comparing to the standard form, that would be y - (-1) = m(x - 2). So that would mean the line is passing through the point (2, -1), right? Because in the point-slope form, the point is (x1, y1). So here, x1 is 2 and y1 is -1. So (2, -1) is a point on the line, regardless of the value of m. But the question is saying that no matter what m is, the line always passes through a fixed point. So if m changes, the line changes, but there's one point that's always on all those lines. So if I think about it, the line is given in a form that already shows that (2, -1) is on the line, because that's the point-slope form. So is that the fixed point? Then the answer would be D: (2, -1). But wait, let me check to make sure. Maybe there's a trick here.Alternatively, maybe the question is trying to test understanding that even though m is varying, the line still passes through some fixed point. Let me verify. Let's consider different values of m and see if (2, -1) is always on the line.Take m = 0. Then the equation becomes y + 1 = 0*(x - 2) => y + 1 = 0 => y = -1. So when m=0, the line is horizontal at y = -1. The point (2, -1) is on this line because when x=2, y=-1. Take m = 1. Then the equation is y + 1 = 1*(x - 2) => y = x - 2 - 1 => y = x - 3. If x=2, then y=2 - 3 = -1. So again, (2, -1) is on the line.Take m = 2. Then y + 1 = 2(x - 2) => y = 2x - 4 - 1 => y = 2x - 5. When x=2, y=4 -5 = -1. So again, (2, -1) is on the line.Hmm, so it seems like regardless of m, when x is 2, y is -1. Therefore, (2, -1) is always on the line. So that's the fixed point, which is option D. But let me check the other options to make sure.Option A is (1, -2). Let's plug x=1 into the equation. Then y + 1 = m(1 - 2) => y + 1 = -m => y = -m -1. So unless m is fixed, y depends on m. For example, if m=0, then y=-1. If m=1, then y=-2. So (1, y) changes with m. Therefore, (1, -2) is only on the line when m=1, but not for other m's. So A is not the fixed point.Option B is (-1, 2). Let's plug x=-1, y=2 into the equation. 2 + 1 = m(-1 - 2) => 3 = m(-3) => m = -1. So this point is only on the line when m=-1, but not for other m's. So B is not the fixed point.Option C is (-2, -1). Plugging x=-2, y=-1 into the equation: -1 + 1 = m(-2 - 2) => 0 = m(-4). So this equation holds only when m=0. So the point is on the line only when m=0, but not for other m's. Hence, C is not the fixed point.Therefore, the only point that is on the line regardless of m is D: (2, -1). Which makes sense because the equation is in the point-slope form around that point. So yeah, I think the answer is D.Wait, but let me think again. Maybe there's another way to approach this problem. Suppose we need to find a point (a, b) that satisfies the equation y +1 = m(x - 2) for any m. So substituting x=a, y=b into the equation:b + 1 = m(a - 2)We need this equation to hold for all values of m. The left side, b + 1, is a constant, and the right side is m(a - 2). For this equation to be true for all m, the coefficient of m must be zero, otherwise, as m changes, the right side changes but the left side remains the same. Therefore, we have two conditions:1. The coefficient of m must be zero: (a - 2) = 0 => a = 22. The constant term must equal the left side: b + 1 = 0 => b = -1Therefore, the fixed point is (2, -1), which is option D. That's another way to see it, by solving for a and b such that the equation holds for all m. So that confirms the answer is D.I think that's solid. Both by plugging in specific values and by solving algebraically, we get that (2, -1) is the fixed point. So the answer must be D.Final Answerboxed{D}</think>The given line equation is ( y + 1 = m(x - 2) ). This is in the point-slope form of a line, where the point ((x_1, y_1)) is ((2, -1)). This means that regardless of the value of ( m ), the line always passes through the point ((2, -1)).To verify, we can substitute ( x = 2 ) and ( y = -1 ) into the equation:[-1 + 1 = m(2 - 2) implies 0 = 0]This equation holds true for any value of ( m ), confirming that ((2, -1)) is a fixed point on the line.Additionally, solving algebraically for a point ((a, b)) that satisfies the equation for all ( m ):[b + 1 = m(a - 2)]For this to hold for all ( m ), the coefficient of ( m ) must be zero, giving ( a - 2 = 0 implies a = 2 ). The constant term must also match, giving ( b + 1 = 0 implies b = -1 ). Thus, the fixed point is ((2, -1)).Therefore, the coordinates of the fixed point are (boxed{D}).,<think>Okay, so I need to figure out which fixed point the line y + 1 = m(x - 2) always passes through, no matter what the real value of m is. The options are given as A: (1, -2), B: (-1, 2), C: (-2, -1), D: (2, -1). Let me think step by step.First, the equation given is y + 1 = m(x - 2). This looks like a point-slope form of a line. The general point-slope form is y - y1 = m(x - x1), where (x1, y1) is a point the line passes through, and m is the slope. In this case, comparing to the general form, the equation can be rewritten as y - (-1) = m(x - 2). So, that suggests that the line passes through the point (2, -1). But wait, the question says that no matter what m is, the line always passes through a fixed point. So is that point (2, -1)? Because in the point-slope form, that point is always on the line regardless of m. But let me verify this.Wait, if m is changing, the line is rotating around that fixed point (2, -1), right? Because different m values give different slopes, but if they all pass through (2, -1), that would make sense. But let me check the answer options. Option D is (2, -1). So maybe D is the correct answer. But let me not jump to conclusions. Let me test each of the options to make sure.The problem states that no matter what m is, the fixed point must satisfy the equation y + 1 = m(x - 2). So, for the coordinates (x, y) to satisfy the equation for all m, the equation must hold true regardless of m. Let's denote the fixed point as (a, b). Then, substituting x = a and y = b into the equation:b + 1 = m(a - 2)This equation must hold true for all values of m. But here, the left side is a constant (since a and b are fixed), and the right side is m times (a - 2). The only way this equation can be true for all m is if the coefficient of m is zero, and the constant term on the left is also zero. That is:a - 2 = 0 (to eliminate the dependence on m)andb + 1 = 0 (to satisfy the remaining constant term)Solving these equations:a = 2b = -1Therefore, the fixed point must be (2, -1), which is option D. Let me confirm by plugging in option D into the equation. Let's substitute x = 2 and y = -1 into y + 1 = m(x - 2):Left side: (-1) + 1 = 0Right side: m(2 - 2) = m*0 = 0So indeed, 0 = 0, which is always true regardless of m. Therefore, the point (2, -1) is on the line for any m. To be thorough, let's check the other options to make sure they don't work. Let's take option A: (1, -2). Substitute x = 1, y = -2 into the equation:Left side: (-2) + 1 = -1Right side: m(1 - 2) = m(-1) = -mSo, -1 = -m. This would require m = 1 for the equation to hold. But since m can be any real number, this isn't valid for all m. Therefore, option A is not correct.Option B: (-1, 2). Substitute x = -1, y = 2:Left side: 2 + 1 = 3Right side: m(-1 - 2) = m(-3) = -3mSo, 3 = -3m → m = -1. Again, this only works for m = -1, not for all m. So option B is out.Option C: (-2, -1). Substitute x = -2, y = -1:Left side: (-1) + 1 = 0Right side: m(-2 - 2) = m(-4) = -4mSo 0 = -4m → m = 0. But if m is not zero, this doesn't hold. Hence, option C is invalid.Thus, only option D works for all values of m. So the answer is D: (2, -1).Final Answerboxed{D}</think>The given line equation is ( y + 1 = m(x - 2) ). This is in the point-slope form ( y - y_1 = m(x - x_1) ), where ((x_1, y_1)) is a fixed point the line passes through. For the equation to hold true for all values of ( m ), the point ((a, b)) must satisfy the equation regardless of ( m ).Substituting ( x = a ) and ( y = b ) into the equation, we get:[b + 1 = m(a - 2)]For this equation to hold for all ( m ), both the coefficient of ( m ) and the constant term must be zero. This gives:[a - 2 = 0 quad text{and} quad b + 1 = 0]Solving these, we find:[a = 2 quad text{and} quad b = -1]Thus, the fixed point is ((2, -1)).Verifying by substituting ((2, -1)) into the original equation:[-1 + 1 = m(2 - 2) implies 0 = 0]This is always true for any ( m ).Checking other options confirms they do not satisfy the equation for all ( m ). Therefore, the correct answer is (boxed{D}).]

🔑:From the given, we set y+1=0 and x-2=0. Under these conditions, the equation is independent of m. Therefore, when x=2 and y=-1, it is independent of m. Thus, the coordinates of the fixed point are (2, -1). Hence, the correct option is boxed{text{D}}.

🧠:[<think>Okay, so I need to figure out which quadrant the complex number z is in after multiplying (2 - i) by (1 - i). Hmm, let me start by recalling that in the complex plane, the real part is the x-axis and the imaginary part is the y-axis. The quadrants are divided based on the signs of the real and imaginary components. First quadrant is both positive, second is negative real and positive imaginary, third is both negative, and fourth is positive real and negative imaginary. First, I need to perform the multiplication of the two complex numbers. Let me write it out step by step. The formula for multiplying two complex numbers (a + bi)(c + di) is ac + adi + bci + bdi². Since i² is -1, the result becomes (ac - bd) + (ad + bc)i. Applying this to (2 - i)(1 - i):Let a = 2, b = -1, c = 1, d = -1.So first, compute the real part: ac - bd. That would be 2*1 - (-1)*(-1). Let me calculate that. 2*1 is 2. Then (-1)*(-1) is 1. So 2 - 1 = 1. Next, compute the imaginary part: ad + bc. That's 2*(-1) + (-1)*1. So that's -2 + (-1) = -3. Therefore, z = 1 - 3i. Now, the real part is 1 (positive) and the imaginary part is -3 (negative). In the complex plane, that would place the point in the fourth quadrant because the real part is positive and the imaginary part is negative. Let me double-check my multiplication to make sure I didn't make a mistake. Multiply (2 - i)(1 - i) using distributive property:First, 2*1 = 2.Then 2*(-i) = -2i.Then (-i)*1 = -i.Then (-i)*(-i) = i² = -1.Adding all these together: 2 - 2i - i -1. Combine like terms: (2 -1) + (-2i -i) = 1 - 3i. Yep, that's correct. So real part 1, positive; imaginary part -3, negative. So quadrant IV, which is the fourth quadrant. Looking at the options, D is the fourth quadrant. Therefore, the answer should be D. Wait, let me just visualize the complex plane to be sure. The first quadrant is top right (positive real, positive imaginary), second is top left (negative real, positive imaginary), third is bottom left (negative real, negative imaginary), and fourth is bottom right (positive real, negative imaginary). So with 1 on the real axis and -3i on the imaginary, the point is 1 to the right and 3 down, which is indeed the fourth quadrant. No mistakes here. Confirming once more, the multiplication steps. (2 - i)(1 - i) = 2*1 + 2*(-i) + (-i)*1 + (-i)*(-i) = 2 - 2i - i + i². Since i² is -1, that becomes 2 - 3i -1 = 1 - 3i. Yep, that's right. Alright, so definitely quadrant IV. So the answer is D.Final Answerboxed{D}</think>To determine the quadrant in which the complex number ( z = (2 - i)(1 - i) ) lies, we start by performing the multiplication.Using the distributive property (FOIL method) for complex numbers:[(2 - i)(1 - i) = 2 cdot 1 + 2 cdot (-i) + (-i) cdot 1 + (-i) cdot (-i)]Calculating each term:[= 2 - 2i - i + i^2]Since ( i^2 = -1 ):[= 2 - 2i - i - 1]Combining like terms:[= (2 - 1) + (-2i - i) = 1 - 3i]The resulting complex number is ( 1 - 3i ), which has a positive real part (1) and a negative imaginary part (-3). In the complex plane, this places the point in the fourth quadrant (positive real, negative imaginary).Thus, the answer is (boxed{D}).,<think>Okay, let me try to figure this out. The problem is asking which quadrant the complex number z is in after multiplying (2 - i) by (1 - i). Hmm, quadrants in the complex plane are similar to the Cartesian coordinate system, right? So the real part is like the x-axis and the imaginary part is like the y-axis. The quadrants are divided based on the signs of the real and imaginary components. First quadrant: Both real and imaginary parts are positive.Second quadrant: Real negative, imaginary positive.Third quadrant: Both real and imaginary parts are negative.Fourth quadrant: Real positive, imaginary negative.So, to find out which quadrant z is in, I need to multiply those two complex numbers and then check the signs of the real and imaginary parts.Let me recall how to multiply complex numbers. It's just like expanding brackets using distributive property. So (a + b)(c + d) = ac + ad + bc + bd. But since we have a minus sign here, I should be careful with the signs.Let's write out the multiplication step by step:(2 - i)(1 - i) Multiply the first terms: 2 * 1 = 2Multiply the outer terms: 2 * (-i) = -2iMultiply the inner terms: (-i) * 1 = -iMultiply the last terms: (-i) * (-i) = i²Now, add all those together:2 - 2i - i + i²Combine like terms. The constants are 2 and i², and the imaginary terms are -2i and -i. Wait, but i² is equal to -1, right? Because i is the imaginary unit where i² = -1.So replacing i² with -1:2 - 1 + (-2i - i)Calculating the real parts: 2 - 1 = 1Calculating the imaginary parts: (-2i - i) = -3iSo putting it all together, z = 1 - 3i.Now, looking at the real and imaginary parts: the real part is 1 (positive) and the imaginary part is -3i (negative). In the complex plane, that would correspond to a point with coordinates (1, -3). Plotting that on the complex plane: the real axis is horizontal, imaginary vertical. Positive real is right, negative real is left. Positive imaginary is up, negative imaginary is down. So (1, -3) is in the fourth quadrant because the real part is positive and the imaginary part is negative.Wait, let me make sure I didn't mix up the quadrants. Quadrant I: (+, +), Quadrant II: (-, +), Quadrant III: (-, -), Quadrant IV: (+, -). Yep, so (1, -3) is in Quadrant IV. So the answer should be D: Fourth quadrant.But just to double-check my multiplication steps in case I made a mistake there. Let me go through it again.(2 - i)(1 - i):First, 2*1 = 22*(-i) = -2i(-i)*1 = -i(-i)*(-i) = i² = -1. That's correct.Adding up: 2 -2i -i -1. Wait, hold on, is that 2 -1 = 1, and -2i -i = -3i. So yeah, 1 -3i. That seems right.So real part is positive, imaginary part is negative. Quadrant IV. Answer D. I think that's correct. Let me think if there's another way to approach this, maybe by plotting the original complex numbers and seeing how multiplication affects them?Original numbers: 2 - i is in the fourth quadrant (real positive, imaginary negative). Similarly, 1 - i is also in the fourth quadrant. But when you multiply two complex numbers, the result's angle is the sum of the angles of the original numbers. The quadrants depend on the sum of those angles.But maybe that's overcomplicating. Since both are in the fourth quadrant, adding their angles... The angle of 2 - i is below the real axis, and the angle of 1 - i is also below the real axis. Adding those two angles would result in an angle further below, but since angles wrap around 360 degrees, adding two negative angles would result in a more negative angle, which would still point into the fourth quadrant? Wait, but if angles are measured from the positive real axis, counterclockwise is positive, clockwise is negative. So 2 - i would have an angle of, let's see, arctangent(-1/2), which is in the fourth quadrant. Similarly, 1 - i has an angle of arctangent(-1/1) = -45 degrees. So adding those two angles: arctangent(-1/2) + (-45 degrees). Hmm, not sure if that necessarily lands in the fourth quadrant. But maybe the magnitudes multiply, and angles add. But perhaps it's easier to just do the multiplication as I did before.Alternatively, perhaps converting to polar form and multiplying? Let's see.First, 2 - i. The modulus is sqrt(2² + (-1)²) = sqrt(4 + 1) = sqrt(5). The angle θ1 is arctangent(-1/2). Since it's in the fourth quadrant, the angle is -arctan(1/2) or equivalently 360 - arctan(1/2) degrees.Second, 1 - i. The modulus is sqrt(1² + (-1)²) = sqrt(2). The angle θ2 is arctangent(-1/1) = -45 degrees or 315 degrees.When you multiply two complex numbers in polar form, you multiply their moduli and add their angles. So modulus of z is sqrt(5)*sqrt(2) = sqrt(10). The angle would be θ1 + θ2 = (-arctan(1/2)) + (-45 degrees). Let's calculate that.First, arctan(1/2) is approximately 26.565 degrees. So θ1 is -26.565 degrees. Then θ2 is -45 degrees. Adding them gives -26.565 -45 = -71.565 degrees. To express this as a positive angle, add 360: 360 -71.565 = 288.435 degrees. Now, 288.435 degrees is in the fourth quadrant, since 270 degrees is the start of the fourth quadrant (270 to 360 degrees). So that confirms it. The angle is in the fourth quadrant. So the result is in the fourth quadrant. Therefore, answer D.Alternatively, maybe this was overcomplicating, but both methods give the same result, which is good. So I'm confident the answer is D.Final Answerboxed{D}</think>To determine which quadrant the complex number ( z = (2 - i)(1 - i) ) lies in, we need to multiply the two complex numbers and then check the signs of the real and imaginary parts.First, we expand the product using the distributive property:[(2 - i)(1 - i) = 2 cdot 1 + 2 cdot (-i) + (-i) cdot 1 + (-i) cdot (-i)]Calculating each term:- ( 2 cdot 1 = 2 )- ( 2 cdot (-i) = -2i )- ( (-i) cdot 1 = -i )- ( (-i) cdot (-i) = i^2 = -1 )Combining all terms:[2 - 2i - i - 1]Simplifying the real and imaginary parts:- Real part: ( 2 - 1 = 1 )- Imaginary part: ( -2i - i = -3i )Thus, the result is ( z = 1 - 3i ). The real part is positive (1) and the imaginary part is negative (-3i), which corresponds to the point (1, -3) in the complex plane. This point is located in the fourth quadrant.Therefore, the answer is (boxed{D}).]

🔑:To solve this problem, we start by expanding the given expression for z:[z = (2-i)(1-i)]We apply the distributive property (also known as FOIL for binomials) to expand:[= 2(1) + 2(-i) - i(1) - i(-i)]Simplifying each term:[= 2 - 2i - i + i^2]Recall that i^2 = -1, so we substitute and simplify:[= 2 - 2i - i - 1]Combining like terms:[= 1 - 3i]Therefore, we have found that z = 1 - 3i. This corresponds to the point (1, -3) in the complex plane. To determine the quadrant, we examine the signs of the real and imaginary parts. The real part is positive (1) and the imaginary part is negative (-3), which places the point in the fourth quadrant.Thus, the correct answer is encapsulated as boxed{D}.