🧠:[<think>Okay, so I need to find the area of triangle ABC with vertices at A(0,0), B(1424233,2848467), and C(1424234,2848469). Hmm, let's see. I remember there's a formula for the area of a triangle when you know the coordinates of the vertices. It's something with determinants or the shoelace formula. Let me recall.The shoelace formula says that the area is half the absolute value of the sum over the edges of x_i y_{i+1} - x_{i+1} y_i. Right? So for three points A, B, C, the area is ½ |x_A(y_B - y_C) + x_B(y_C - y_A) + x_C(y_A - y_B)|. Let me write that down properly.Given points A(x1, y1), B(x2, y2), C(x3, y3), the area is:½ |x1(y2 - y3) + x2(y3 - y1) + x3(y1 - y2)|So plugging in the coordinates here:A is (0,0), so x1 = 0, y1 = 0B is (1424233,2848467), so x2 = 1424233, y2 = 2848467C is (1424234,2848469), so x3 = 1424234, y3 = 2848469Plugging into the formula:½ |0*(2848467 - 2848469) + 1424233*(2848469 - 0) + 1424234*(0 - 2848467)|Simplify term by term:First term: 0*(something) = 0Second term: 1424233*(2848469) = let's calculate thatThird term: 1424234*(-2848467) = let's calculate thatSo the area is ½ |0 + (1424233*2848469) - (1424234*2848467)|Hmm, that looks like it might be a small number because the two products are similar but subtracted. Let me compute the difference between the two products.Let me denote:Let me compute 1424233*2848469 - 1424234*2848467Notice that 1424234 = 1424233 + 1 and 2848469 = 2848467 + 2Wait, maybe factor this expression.Alternatively, let's compute the difference:1424233*2848469 - 1424234*2848467= 1424233*(2848467 + 2) - (1424233 + 1)*2848467= 1424233*2848467 + 1424233*2 - 1424233*2848467 - 1*2848467Simplify:The 1424233*2848467 terms cancel out, leaving:1424233*2 - 2848467= 2848466 - 2848467= -1So the entire expression inside the absolute value is |-1| = 1, so the area is ½ * 1 = 0.5Wait, that's interesting. So despite the coordinates being huge numbers, the area is 0.5? Let me verify this step by step to make sure I didn't make a mistake.First, confirm the coordinates:A(0,0), B(1424233,2848467), C(1424234,2848469)Calculating the area using shoelace formula:Compute:0*(2848467 - 2848469) = 0*( -2 ) = 01424233*(2848469 - 0) = 1424233*28484691424234*(0 - 2848467) = 1424234*(-2848467)So the sum is 1424233*2848469 - 1424234*2848467Let me check the algebra here:Let me write 1424233*2848469 as (1424234 - 1)*(2848467 + 2)Expanding this:= 1424234*2848467 + 1424234*2 - 1*2848467 - 1*2= 1424234*2848467 + 2848468 - 2848467 - 2= 1424234*2848467 + (2848468 - 2848467) - 2= 1424234*2848467 + 1 - 2= 1424234*2848467 - 1Therefore, the expression 1424233*2848469 - 1424234*2848467 becomes:(1424234*2848467 - 1) - 1424234*2848467 = -1So yes, the difference is -1, absolute value is 1, so area is ½ *1 = 0.5So the area is 0.5, which is 0.50 when rounded to hundredths.But wait, maybe there's something wrong here? Because the coordinates are very large, but the area is 0.5, which seems counterintuitive. Maybe the points are colinear? Wait, but the area is 0.5, which is very small. Wait, but let's check if the three points are colinear.If three points are colinear, the area would be zero. Since the area here is 0.5, they are not colinear, but the area is still very small given the size of the coordinates. Let's check the vectors.Vector AB is B - A = (1424233, 2848467)Vector AC is C - A = (1424234, 2848469)Calculate the determinant of these two vectors to find the area (which is half the absolute value of the determinant):|AB x AC| = |1424233*2848469 - 1424234*2848467|Which is the same expression as before. So same result, which we found was 1. Thus area is 0.5. So this checks out.But just to be thorough, maybe check the slopes.Slope of AB: (2848467 - 0)/(1424233 - 0) = 2848467 / 1424233 ≈ 2. Exactly, 2848467 divided by 1424233. Let me compute that:1424233 * 2 = 2848466. So 2848467 is 1424233*2 + 1. Therefore, 2848467 / 1424233 = 2 + 1/1424233 ≈ 2.0000007Similarly, slope of AC: (2848469 - 0)/(1424234 - 0) = 2848469 / 1424234Similarly, 1424234*2 = 2848468, so 2848469 is 1424234*2 +1, so 2848469/1424234 = 2 + 1/1424234 ≈ 2.0000007Wait, so the slopes are almost the same but differ by 1/1424233 vs 1/1424234. So they are not exactly the same, so the lines AB and AC are not parallel, and the points are not colinear, which is why the area is 0.5.Therefore, after all checks, the area is indeed 0.5, which is 0.50 when rounded to the nearest hundredth.Final AnswerThe area of triangle (ABC) is boxed{0.50}.</think>To find the area of triangle (ABC) with vertices (A(0,0)), (B(1424233, 2848467)), and (C(1424234, 2848469)), we use the shoelace formula. The formula for the area of a triangle given vertices ((x_1, y_1)), ((x_2, y_2)), and ((x_3, y_3)) is:[text{Area} = frac{1}{2} left| x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) right|]Substituting the coordinates of points (A), (B), and (C):[text{Area} = frac{1}{2} left| 0(2848467 - 2848469) + 1424233(2848469 - 0) + 1424234(0 - 2848467) right|]Simplifying the terms inside the absolute value:[= frac{1}{2} left| 0 + 1424233 cdot 2848469 - 1424234 cdot 2848467 right|]We compute the difference between the products:[1424233 cdot 2848469 - 1424234 cdot 2848467]Expanding the terms:[= (1424233)(2848467 + 2) - (1424233 + 1)(2848467)][= 1424233 cdot 2848467 + 1424233 cdot 2 - 1424233 cdot 2848467 - 2848467][= 2848466 - 2848467][= -1]Taking the absolute value and multiplying by (frac{1}{2}):[text{Area} = frac{1}{2} left| -1 right| = frac{1}{2} cdot 1 = 0.5]Rounding to the hundredths place, the area of triangle (ABC) is (boxed{0.50}).,<think>Okay, so I need to find the area of triangle ABC with vertices at A(0,0), B(1424233,2848467), and C(1424234,2848469). Hmm, coordinates are given, so maybe I can use the shoelace formula? Let me recall how that works. The formula is something like taking the sum of products of coordinates in one diagonal direction minus the sum in the other diagonal direction, then taking half the absolute value of that. Let me write down the coordinates:A is (0, 0)B is (1424233, 2848467)C is (1424234, 2848469)So, applying the shoelace formula, the area should be (1/2)|x1(y2 - y3) + x2(y3 - y1) + x3(y1 - y2)|. Let me plug in the values here.First, x1 is 0, y1 is 0. Then x2 is 1424233, y2 is 2848467. x3 is 1424234, y3 is 2848469.So substituting into the formula:Area = (1/2)|0*(2848467 - 2848469) + 1424233*(2848469 - 0) + 1424234*(0 - 2848467)|Let me compute each term step by step.First term: 0*(2848467 - 2848469) = 0*( -2 ) = 0.Second term: 1424233*(2848469 - 0) = 1424233 * 2848469. That's a big number. Maybe I can compute this more cleverly.Third term: 1424234*(0 - 2848467) = 1424234*(-2848467) = -1424234*2848467.So putting it all together:Area = (1/2)|0 + 1424233*2848469 - 1424234*2848467|Hmm, notice that 1424233*2848469 and 1424234*2848467 are very similar. Let me see if I can factor something out here.Let me denote x = 1424233 and y = 2848467. Then the two terms become x*(y + 2) and (x + 1)*y. Let's check:Wait, original points:B is (x, y) = (1424233, 2848467)C is (x + 1, y + 2) = (1424234, 2848469)So, indeed, the second term is x*(y + 2) and the third term is (x + 1)*y. So then:Area = (1/2)|x(y + 2) - (x + 1)y|Let's compute that expression inside the absolute value:x(y + 2) - (x + 1)y = xy + 2x - xy - y = 2x - yTherefore, the area becomes (1/2)|2x - y|.So substituting x and y back:2x = 2*1424233 = 2848466y = 2848467Thus, 2x - y = 2848466 - 2848467 = -1Taking absolute value, |-1| = 1Therefore, area = (1/2)*1 = 0.5Wait, that seems surprisingly simple. Let me verify the steps again to make sure I didn't make a mistake.Starting with the shoelace formula:Area = (1/2)|x1(y2 - y3) + x2(y3 - y1) + x3(y1 - y2)|Plugging in the coordinates:x1 = 0, y1 = 0; x2 = 1424233, y2 = 2848467; x3 = 1424234, y3 = 2848469So,0*(2848467 - 2848469) = 0*(-2) = 01424233*(2848469 - 0) = 1424233*28484691424234*(0 - 2848467) = 1424234*(-2848467)Then, combining these:0 + 1424233*2848469 - 1424234*2848467Which is 1424233*2848469 - 1424234*2848467As I did before, factoring:Let x = 1424233, so 1424234 = x + 1Let y = 2848467, so 2848469 = y + 2So the expression becomes x*(y + 2) - (x + 1)*yExpanding:xy + 2x - xy - y = 2x - yWhich is 2*1424233 - 2848467Compute 2*1424233:1424233 * 2 = 2848466Subtract 2848467: 2848466 - 2848467 = -1Absolute value of -1 is 1, so area is (1/2)*1 = 0.5So the area is 0.5, which when rounded to the hundredths is 0.50. Wait, but 0.5 is already exact to the hundredths place, so it would be 0.50.But let me check another way. Maybe using vectors or determinant.The area can also be calculated using the cross product formula. The area is (1/2)|AB × AC|, where AB and AC are vectors from point A.Vector AB is (1424233, 2848467)Vector AC is (1424234, 2848469)The cross product in 2D is scalar magnitude equal to (x1*y2 - x2*y1)So AB × AC = (1424233*2848469 - 1424234*2848467)Which is the same as before, which computes to -1. So magnitude is 1, area is 0.5. So same result.Therefore, the area is 0.5, which is 0.50 when rounded to the hundredths. So the answer should be 0.50.But wait, maybe the problem expects units? The coordinates are given as numbers, but no units specified, so the area is in square units. Since the coordinates are integers, the area is exactly 0.5, so when rounded to hundredths, it's 0.50.Alternatively, perhaps the problem is a trick question, because points A, B, C are colinear? Wait, but we computed the area as 0.5, which is non-zero, so they are not colinear. Wait, but let's check.If three points are colinear, the area would be zero. Here, we have 0.5, so they are not colinear. Let me check the slopes.Slope of AB: (2848467 - 0)/(1424233 - 0) = 2848467 / 1424233. Let me compute that. 2848467 divided by 1424233. Let's see: 1424233 * 2 = 2848466. So 2848467 / 1424233 = 2 + 1/1424233 ≈ 2.0000007.Slope of AC: (2848469 - 0)/(1424234 - 0) = 2848469 / 1424234. Let me compute that. 1424234 * 2 = 2848468. So 2848469 / 1424234 = 2 + 1/1424234 ≈ 2.0000007.Wait, the slopes are slightly different. Because 1/1424233 versus 1/1424234. So the slopes are not exactly equal, so the points are not colinear, hence area is non-zero, 0.5. That makes sense.Alternatively, since points B and C are very close to each other relative to point A? Wait, the coordinates of B and C differ by (1, 2). So the displacement from B to C is (1, 2). And the displacement from A to B is (1424233, 2848467). So, if we think of vectors, AB is a vector, and AC is another vector.But the cross product between AB and AC is 1, so the area is 0.5. So that's consistent.Alternatively, using matrices or other methods, but I think the shoelace formula and cross product both confirm the area is 0.5.But just to double-check, let me compute AB × AC manually.AB × AC = (1424233)(2848469) - (1424234)(2848467)Let me compute each product:First product: 1424233 * 2848469Second product: 1424234 * 2848467Let me see if there's a way to compute the difference between these two products.Let me write the first product as (1424234 - 1) * (2848467 + 2) = (1424234 - 1)(2848467 + 2)Expanding this:1424234*2848467 + 1424234*2 - 1*2848467 - 1*2Similarly, the second product is 1424234*2848467Therefore, the difference between first and second product is:[1424234*2848467 + 1424234*2 - 2848467 - 2] - 1424234*2848467 =1424234*2 - 2848467 - 2Compute 1424234*2: 2848468Then subtract 2848467: 2848468 - 2848467 = 1Subtract 2: 1 - 2 = -1Therefore, the difference between the two products is -1, so the cross product is -1, absolute value 1, area 0.5. So that's correct.Therefore, after all these checks, the area is indeed 0.5, which is 0.50 when rounded to the hundredths place. So the answer should be 0.50.Final Answerboxed{0.50}</think>To find the area of triangle (ABC) with vertices (A(0,0)), (B(1424233, 2848467)), and (C(1424234, 2848469)), we use the shoelace formula. The formula for the area of a triangle given vertices ((x_1, y_1)), ((x_2, y_2)), and ((x_3, y_3)) is:[text{Area} = frac{1}{2} left| x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) right|]Substituting the coordinates (A(0,0)), (B(1424233, 2848467)), and (C(1424234, 2848469)):[text{Area} = frac{1}{2} left| 0(2848467 - 2848469) + 1424233(2848469 - 0) + 1424234(0 - 2848467) right|]Simplifying each term:[0 cdot (2848467 - 2848469) = 0][1424233 cdot 2848469 = 1424233 cdot 2848469][1424234 cdot (-2848467) = -1424234 cdot 2848467]Combining these terms:[text{Area} = frac{1}{2} left| 1424233 cdot 2848469 - 1424234 cdot 2848467 right|]Let (x = 1424233) and (y = 2848467). Then the expression inside the absolute value becomes:[x(y + 2) - (x + 1)y = xy + 2x - xy - y = 2x - y]Substituting back:[2x = 2 cdot 1424233 = 2848466][2x - y = 2848466 - 2848467 = -1]Taking the absolute value:[left| -1 right| = 1]Thus, the area is:[text{Area} = frac{1}{2} cdot 1 = 0.5]Rounding to the hundredths place, the area is (0.50).[boxed{0.50}]]

🔑:Solution. Note that points B and C lie on the line y=2x+1. Their abscissas differ by 1, hence BC=sqrt{5}. The length of the height of triangle ABC, drawn from vertex A, is equal to the distance h from point A to the line y=2x+1, which, in turn, is equal to 1 / sqrt{5}. The desired areaS_{ABC}=frac{1}{2} cdot h cdot BC=frac{1}{2}Answer: 0.50.

🧠:[<think>Okay, let's see. I have this problem where I need to find the real number "a" such that the curves y = (1/(2e))x² and y = a ln x have a common tangent at their common point P(s, t). Hmm, so first, I need to recall what it means for two curves to have a common tangent at a point. A common tangent means that not only do the two curves pass through the same point P(s, t), but also their slopes (derivatives) at that point are equal. So, basically, there are two conditions here: 1. The point (s, t) lies on both curves. Therefore, substituting s into both equations should give the same t-value.2. The derivatives of both curves at x = s are equal. Alright, let me write that down step by step. First, let's denote the two functions as f(x) = (1/(2e))x² and g(x) = a ln x. Condition 1: Both functions pass through (s, t). So:f(s) = g(s) => (1/(2e))s² = a ln s. Condition 2: Their derivatives at x = s are equal. Let's compute the derivatives.f'(x) = d/dx [ (1/(2e))x² ] = (1/(2e)) * 2x = x/e. So, f'(s) = s/e.g'(x) = d/dx [ a ln x ] = a*(1/x). Therefore, g'(s) = a/s. Setting them equal:f'(s) = g'(s) => s/e = a/s. So, from the second condition, we can solve for a in terms of s. Let's do that first.From s/e = a/s, multiply both sides by s:s²/e = a Therefore, a = s²/e. Now, let's look back at the first condition: (1/(2e))s² = a ln s. But we just found that a = s²/e. So, substitute that into the first equation:(1/(2e))s² = (s²/e) ln s Hmm, okay. Let's write that equation out:(1/(2e))s² = (s²/e) ln s First, I can multiply both sides by e to eliminate the denominators:(1/2)s² = s² ln s Subtract (1/2)s² from both sides:0 = s² ln s - (1/2)s² Factor out s²:s² (ln s - 1/2) = 0 So, this gives two possibilities: either s² = 0 or ln s - 1/2 = 0. But s² = 0 implies s = 0. However, the second function is a ln x, which is undefined at x = 0. Therefore, s cannot be 0. Therefore, the other factor must be zero: ln s - 1/2 = 0 Solving for s:ln s = 1/2 Exponentiating both sides:s = e^(1/2) = sqrt(e) Okay, so s is sqrt(e). Now, let's find a. Earlier, we found that a = s²/e. So plugging s = sqrt(e):a = (sqrt(e))² / e = e / e = 1 So, a is 1. Let me verify this. First, check if s = sqrt(e) is a common point. Let's compute f(s):f(sqrt(e)) = (1/(2e))*(sqrt(e))² = (1/(2e))*e = 1/2 g(sqrt(e)) = a ln(sqrt(e)) = a*(1/2) ln e = a*(1/2)*1 = a/2 Since we said a = 1, then g(sqrt(e)) = 1/2, which matches f(sqrt(e)). So, that's correct. Now check the derivatives at s = sqrt(e):f'(sqrt(e)) = sqrt(e)/e = 1/sqrt(e) g'(sqrt(e)) = a/sqrt(e) Since a = 1, this is also 1/sqrt(e). So the derivatives are equal. Therefore, everything checks out. So the real number a is 1. I think that's it. Let me just recap to make sure I didn't skip any steps. We set up the two conditions for the common tangent: the points must be the same, and the derivatives must be equal. Then we solved the system of equations, found s in terms of e, substituted back into the expression for a, and verified the result. Seems solid. Wait, just to be thorough, let me check the equations again. The key step was when I substituted a = s²/e into the first equation. Let me verify that step again. From the first equation:(1/(2e))s² = a ln s But a is s²/e, so:(1/(2e))s² = (s²/e) ln s Multiply both sides by e:(1/2)s² = s² ln s Divide both sides by s² (since s ≠ 0):1/2 = ln s Therefore, ln s = 1/2, so s = e^{1/2}. Exactly. So that step is correct. Therefore, everything follows, and a is indeed 1. Final AnswerThe real number ( a ) is boxed{1}.</think>Given the curves ( y = frac{1}{2e}x^2 ) and ( y = a ln x ), we need to find the real number ( a ) such that they have a common tangent at their common point ( P(s, t) ).First, we note that for the curves to have a common tangent at ( P(s, t) ), both the points and their derivatives must be equal at ( x = s ).1. Condition for the point ( P(s, t) ): [ frac{1}{2e}s^2 = a ln s ]2. Condition for the derivatives at ( x = s ): [ text{Derivative of } frac{1}{2e}x^2 text{ is } frac{x}{e} implies frac{s}{e} ] [ text{Derivative of } a ln x text{ is } frac{a}{x} implies frac{a}{s} ] Setting the derivatives equal: [ frac{s}{e} = frac{a}{s} ] Solving for ( a ): [ a = frac{s^2}{e} ]Substituting ( a = frac{s^2}{e} ) into the first condition: [ frac{1}{2e}s^2 = frac{s^2}{e} ln s ] Simplifying: [ frac{1}{2} = ln s ] Solving for ( s ): [ ln s = frac{1}{2} implies s = e^{1/2} = sqrt{e} ]Substituting ( s = sqrt{e} ) back into ( a = frac{s^2}{e} ): [ a = frac{(sqrt{e})^2}{e} = frac{e}{e} = 1 ]Thus, the real number ( a ) is (boxed{1}).,<think>Okay, so I have this problem here where I need to find the real number "a" such that the curves y = (1/(2e))x² and y = a ln x have a common tangent at their common point P(s, t). Hmm, let me start by understanding what this means.First, a common tangent means that at the point P(s, t), both curves not only intersect but also have the same slope. So, there are two conditions here: the points must be the same (i.e., both curves pass through (s, t)), and their derivatives at that point must be equal. That makes sense. So, I need to set up equations based on these two conditions and solve for "a".Let me start by writing down the equations for the two curves:1. y = (1/(2e))x²2. y = a ln xSince they have a common point P(s, t), substituting x = s into both equations should give the same y-value, t. So:From the first curve: t = (1/(2e))s²From the second curve: t = a ln sTherefore, I can set these two expressions for t equal to each other:(1/(2e))s² = a ln s ---> Equation (1)Now, the second condition is that the derivatives at x = s must be equal. Let's compute the derivatives of both curves.For the first curve, dy/dx = d/dx [(1/(2e))x²] = (1/(2e))*2x = x/e. So at x = s, the slope is s/e.For the second curve, dy/dx = d/dx [a ln x] = a*(1/x). So at x = s, the slope is a/s.Setting these equal gives:s/e = a/s ---> Equation (2)Now, we have two equations: Equation (1) and Equation (2), and two unknowns: a and s. So, we can solve this system of equations.From Equation (2):s/e = a/sMultiply both sides by s:s²/e = aTherefore, a = s²/e ---> Equation (2a)Now, substitute this into Equation (1):(1/(2e))s² = (s²/e) ln sHmm, okay. Let me write that again:Left side: (1/(2e))s²Right side: (s²/e) ln sSo, let's set them equal:(1/(2e))s² = (s²/e) ln sFirst, I can divide both sides by s² (assuming s ≠ 0, which is reasonable here because ln x is only defined for x > 0, so s must be positive). Then:1/(2e) = (1/e) ln sMultiply both sides by e to eliminate denominators:1/2 = ln sTherefore:ln s = 1/2Exponentiating both sides gives:s = e^(1/2) = sqrt(e)So, s = sqrt(e). Then, from Equation (2a):a = s² / eBut s² is (sqrt(e))² = e. Therefore:a = e / e = 1So, a equals 1. Let me verify this to make sure I didn't make any mistakes.First, check if s = sqrt(e) is valid. Since sqrt(e) is positive, yes. Then, compute t from both curves.From the first curve: t = (1/(2e))(sqrt(e))² = (1/(2e))*e = 1/2From the second curve: t = a ln s. If a = 1, then t = ln(sqrt(e)) = ln(e^(1/2)) = (1/2) ln e = 1/2*1 = 1/2. So that checks out.Now check the derivatives. From the first curve: dy/dx at x = sqrt(e) is sqrt(e)/e = 1/sqrt(e).From the second curve: dy/dx at x = sqrt(e) is a/s = 1/sqrt(e). Which matches. So both the y-values and the derivatives agree. Therefore, a = 1 is correct.I think that's the answer. Let me just recap the steps to ensure clarity:1. Found the expressions for t from both curves and set them equal.2. Took derivatives of both curves and set them equal at x = s.3. Solved the system of equations to find a in terms of s, then found s by substitution.4. Plugged back to find a and verified the solution.Everything seems consistent. So, the real number a is 1.Final AnswerThe real number (a) is boxed{1}.</think>To find the real number (a) such that the curves (y = frac{1}{2e}x^2) and (y = a ln x) have a common tangent at their common point (P(s, t)), we need to satisfy two conditions: the curves must intersect at ((s, t)) and their derivatives at that point must be equal.1. Intersection Condition: - For the first curve: (t = frac{1}{2e}s^2) - For the second curve: (t = a ln s) - Equating the two expressions for (t): [ frac{1}{2e}s^2 = a ln s quad text{(Equation 1)} ]2. Slope Condition: - Derivative of the first curve: (frac{dy}{dx} = frac{x}{e}). At (x = s), the slope is (frac{s}{e}). - Derivative of the second curve: (frac{dy}{dx} = frac{a}{x}). At (x = s), the slope is (frac{a}{s}). - Equating the slopes: [ frac{s}{e} = frac{a}{s} quad text{(Equation 2)} ]From Equation 2, solving for (a):[s^2 = ae implies a = frac{s^2}{e} quad text{(Equation 2a)}]Substituting (a = frac{s^2}{e}) into Equation 1:[frac{1}{2e}s^2 = frac{s^2}{e} ln s]Dividing both sides by (frac{s^2}{e}):[frac{1}{2} = ln s]Solving for (s):[ln s = frac{1}{2} implies s = e^{1/2} = sqrt{e}]Substituting (s = sqrt{e}) back into Equation 2a:[a = frac{(sqrt{e})^2}{e} = frac{e}{e} = 1]Verification:- For the first curve: (t = frac{1}{2e}(sqrt{e})^2 = frac{1}{2})- For the second curve: (t = 1 cdot ln sqrt{e} = frac{1}{2})- Slopes: Both are (frac{1}{sqrt{e}})Thus, the real number (a) is (boxed{1}).]

🔑:First, let's find the derivative of the curve y= frac {1}{2e}x^2:[y' = frac{x}{e}]The slope of the tangent at point P(s, t) is then:[k = frac{s}{e}]Next, find the derivative of the curve y=alnx:[y' = frac{a}{x}]The slope of the tangent at point P(s, t) is:[k = frac{a}{s}]Since the curves have a common tangent at the point P(s, t), their slopes must be equal at that point:[frac{s}{e} = frac{a}{s}]Also, since P(s, t) is a common point on both curves, the y-coordinates must be the same for both equations:[t = frac {1}{2e}s^2][t = alns]Now we can set up a system of equations to solve for s and a:[begin{cases}s/e = a/s (1/2e)s^2 = alnsend{cases}]From the first equation, crossing multiplying gives us s^2 = ae. Substitute s^2 = ae into the second equation:[(1/2e)(ae) = alns][a/2 = alns]Divide both sides by a (assuming a neq 0):[1/2 = lns]Take the exponent of both sides:[e^{1/2} = s][s^2 = e]Now we know that s^2 = ae and s^2 = e, so by comparing these, we can directly find the value of a:[a = boxed{1}]

🧠:[<think>Okay, so I need to find the value of k in the quartic equation x⁴ - 18x³ + kx² + 200x - 1984 = 0, given that the product of two of its roots is -32. Hmm, quartic equations can be tricky, but maybe I can use Vieta's formulas here. Vieta's relates the coefficients of a polynomial to sums and products of its roots. Let me recall Vieta's formulas for a quartic equation.For a quartic equation x⁴ + a x³ + b x² + c x + d = 0, the roots r1, r2, r3, r4 satisfy:- Sum of roots: r1 + r2 + r3 + r4 = -a- Sum of products of roots two at a time: r1r2 + r1r3 + r1r4 + r2r3 + r2r4 + r3r4 = b- Sum of products three at a time: r1r2r3 + r1r2r4 + r1r3r4 + r2r3r4 = -c- Product of roots: r1r2r3r4 = dBut in the given equation, the coefficients are different. Let me write down the equation again:x⁴ - 18x³ + kx² + 200x - 1984 = 0Comparing this to the standard form x⁴ + a x³ + b x² + c x + d = 0, we have:a = -18, b = k, c = 200, d = -1984So applying Vieta's formulas:1. r1 + r2 + r3 + r4 = -a = 182. r1r2 + r1r3 + r1r4 + r2r3 + r2r4 + r3r4 = b = k3. r1r2r3 + r1r2r4 + r1r3r4 + r2r3r4 = -c = -2004. r1r2r3r4 = d = -1984Now, the problem states that the product of two of the roots is -32. Let's say those two roots are r1 and r2. So, r1r2 = -32.We need to find k, which is the sum of all products of roots two at a time. So, k = r1r2 + r1r3 + r1r4 + r2r3 + r2r4 + r3r4We know that r1r2 = -32, so if I can find the sum of the other products: r1r3 + r1r4 + r2r3 + r2r4 + r3r4, then adding -32 will give k.Alternatively, maybe I can express other Vieta's formulas in terms of known quantities.We have the product of all four roots: r1r2r3r4 = -1984. Since r1r2 = -32, then r3r4 = (-1984)/(r1r2) = (-1984)/(-32) = 62. So, r3r4 = 62.Also, the sum of all roots is 18: r1 + r2 + r3 + r4 = 18. Let me denote S = r1 + r2 and T = r3 + r4. Then S + T = 18.Additionally, the sum of products three at a time is -200. Let's write that out:r1r2r3 + r1r2r4 + r1r3r4 + r2r3r4 = -200Factorizing terms:r1r2(r3 + r4) + r3r4(r1 + r2) = -200We know r1r2 = -32 and r3r4 = 62. Let's substitute those in:-32*(r3 + r4) + 62*(r1 + r2) = -200But from S + T = 18, we have r3 + r4 = T = 18 - S. So substituting that in:-32*(18 - S) + 62*S = -200Let me compute that:First, expand the terms:-32*18 + 32*S + 62*S = -200Calculate -32*18:-32*18 = -576Then combine like terms:32S + 62S = 94SSo the equation becomes:-576 + 94S = -200Now, solve for S:94S = -200 + 576 = 376Therefore, S = 376 / 94Divide numerator and denominator by 2: 188 / 47. Hmm, 47*4 = 188, so S = 4Wait, 47*4 = 188? Let me check:47*4: 40*4=160, 7*4=28, so 160+28=188. Yes. So 188/47=4. So S = 4.Therefore, r1 + r2 = S = 4, and r3 + r4 = T = 18 - 4 = 14.So, now we have:- Sum of r1 and r2 is 4- Product of r1 and r2 is -32- Sum of r3 and r4 is 14- Product of r3 and r4 is 62So, maybe r1 and r2 are roots of the quadratic equation x² - Sx + P = 0, where S=4, P=-32. So x² -4x -32 = 0Similarly, r3 and r4 are roots of x² -14x +62 =0Let me check if these quadratics can be factored or have integer roots.First quadratic: x² -4x -32. Let's try factoring. Looking for two numbers that multiply to -32 and add to -4. Hmm, (-8 and 4): -8*4 = -32, -8 +4 = -4. So factors: (x -8)(x +4). Therefore, roots are 8 and -4.So r1 and r2 could be 8 and -4. Their sum is 8 + (-4) = 4, product is -32. Perfect.Second quadratic: x² -14x +62. Let's check discriminant: 14² -4*1*62 = 196 -248 = -52. Negative discriminant, so complex roots. Wait, but the original quartic equation has real coefficients, so complex roots come in conjugate pairs. But if two roots are real (8 and -4) and the other two are complex conjugates, that's possible. But then the product of the complex roots would be 62, which is positive. But since they are conjugates, their product is the square of their modulus, so that's okay.Alternatively, maybe all roots are real. But given that the quadratic for r3 and r4 has a negative discriminant, that can't be. So, perhaps two real roots and two complex roots. But the problem statement doesn't specify whether the roots are real or not. So it's okay.Anyway, the nature of the roots might not matter here. Let's proceed.We need to find k, which is the sum of all products of two roots. We already know r1r2 = -32 and r3r4=62. Then the sum of the other products: r1r3 + r1r4 + r2r3 + r2r4. Let me compute that.Let me denote A = r1 + r2 = 4, B = r3 + r4 =14, C = r1r2 = -32, D = r3r4 =62Then the sum r1r3 + r1r4 + r2r3 + r2r4 can be written as r1(r3 + r4) + r2(r3 + r4) = (r1 + r2)(r3 + r4) = A*BTherefore, this sum is 4*14 =56Therefore, the total sum for k is r1r2 + (r1r3 + r1r4 + r2r3 + r2r4) + r3r4 = C + A*B + DWait, wait. Wait, Vieta's formula for the sum of products two at a time is:r1r2 + r1r3 + r1r4 + r2r3 + r2r4 + r3r4 = kSo we have k = (r1r2) + (r3r4) + (r1r3 + r1r4 + r2r3 + r2r4)Which is equal to C + D + ABSince r1r3 + r1r4 + r2r3 + r2r4 = (r1 + r2)(r3 + r4) = ABSo k = C + D + AB = (-32) + 62 + (4*14) = (-32 +62) +56= 30 +56=86Therefore, k=86.Wait, let me verify this step by step.We have:k = sum of all products two at a time.We know two products: r1r2 = -32 and r3r4=62. The remaining four products are r1r3, r1r4, r2r3, r2r4. These can be grouped as (r1 + r2)(r3 + r4). Which is 4*14=56. Therefore, total k is -32 +62 +56=86. That's correct.But let me check with the equations again.Alternatively, we can use the formula for k, which is:k = r1r2 + r1r3 + r1r4 + r2r3 + r2r4 + r3r4We know r1r2=-32, r3r4=62. The other terms are (r1 + r2)(r3 + r4) =4*14=56. So total k= -32 +62 +56=86.Alternatively, another way to check: Let's compute all the products.Suppose the roots are 8, -4, and the roots of x² -14x +62=0. Let's call them α and β.So, r1=8, r2=-4, r3=α, r4=β.Compute the sum of products two at a time:r1r2 =8*(-4)= -32r1r3=8α, r1r4=8β, r2r3=-4α, r2r4=-4β, r3r4=62So sum = -32 +8α +8β -4α -4β +62Simplify the terms with α and β:8α -4α =4α8β -4β=4βSo total sum = -32 +4(α + β) +62But α + β =14 (from quadratic equation x² -14x +62=0). So:sum = -32 +4*14 +62 = -32 +56 +62= (-32 +56)=24; 24 +62=86. Yep, same result. So k=86.Therefore, the answer is 86. Let me confirm if all Vieta's formulas hold with this value.Given k=86, then the quartic equation is x⁴ -18x³ +86x² +200x -1984=0Check if the product of roots is indeed -1984. The product is r1r2r3r4= (-32)*62= -1984. Correct.Sum of roots:8 + (-4) + α + β=4 +14=18. Correct.Sum of products three at a time: r1r2r3 + r1r2r4 + r1r3r4 + r2r3r4= r1r2(r3 + r4) + r3r4(r1 + r2)= (-32)*14 +62*4= -448 +248= -200. Which matches the coefficient. Correct.Therefore, all Vieta's formulas are satisfied. Thus, k=86 is the correct answer.Final AnswerThe value of ( k ) is (boxed{86}).</think>Given the quartic equation (x^4 - 18x^3 + kx^2 + 200x - 1984 = 0) with the product of two of its roots being (-32), we need to determine the value of (k).Using Vieta's formulas for the roots (r_1, r_2, r_3, r_4) of the quartic equation:1. The sum of the roots: (r_1 + r_2 + r_3 + r_4 = 18)2. The sum of the products of the roots taken two at a time: (r_1r_2 + r_1r_3 + r_1r_4 + r_2r_3 + r_2r_4 + r_3r_4 = k)3. The sum of the products of the roots taken three at a time: (r_1r_2r_3 + r_1r_2r_4 + r_1r_3r_4 + r_2r_3r_4 = -200)4. The product of the roots: (r_1r_2r_3r_4 = -1984)Given (r_1r_2 = -32), we find (r_3r_4 = frac{-1984}{-32} = 62).Let (S = r_1 + r_2) and (T = r_3 + r_4). From the sum of the roots, we have (S + T = 18). Using the sum of the products taken three at a time:[r_1r_2(r_3 + r_4) + r_3r_4(r_1 + r_2) = -200]Substituting the known values:[-32T + 62S = -200]Since (T = 18 - S), substituting (T) gives:[-32(18 - S) + 62S = -200]Solving for (S):[-576 + 32S + 62S = -200 implies 94S = 376 implies S = 4]Thus, (T = 18 - 4 = 14).The sum of the products of the roots taken two at a time is:[r_1r_2 + r_3r_4 + (r_1 + r_2)(r_3 + r_4)]Substituting the known values:[-32 + 62 + 4 times 14 = -32 + 62 + 56 = 86]Thus, the value of (k) is (boxed{86}).,<think>Okay, so I have this quartic equation: x⁴ - 18x³ + kx² + 200x - 1984 = 0. They tell me that the product of two of the four roots is -32, and I need to find the value of k. Hmm, quartic equations can be a bit tricky, but maybe I can use Vieta's formulas here. Let me recall Vieta's formulas for a quartic equation. For a general quartic equation x⁴ + a x³ + b x² + c x + d = 0, the sum of the roots is -a, the sum of the products of the roots two at a time is b, the sum of the products three at a time is -c, and the product of all roots is d. Wait, but in our equation, the coefficients are a bit different. Let me check:Given equation: x⁴ - 18x³ + kx² + 200x - 1984 = 0. So comparing with the general form, the coefficients are:a = -18, b = k, c = 200, d = -1984.Therefore, Vieta's formulas give:Sum of roots: r1 + r2 + r3 + r4 = -a = 18.Sum of products two at a time: r1r2 + r1r3 + r1r4 + r2r3 + r2r4 + r3r4 = b = k.Sum of products three at a time: r1r2r3 + r1r2r4 + r1r3r4 + r2r3r4 = -c = -200.Product of all roots: r1r2r3r4 = d = -1984.Okay, so they told me that the product of two of the roots is -32. Let's say r1*r2 = -32. Then I need to find k, which is the sum of all products two at a time. So maybe if I can express the other products in terms of known quantities?First, let me note that if r1*r2 = -32, then perhaps r3*r4 can be found since the product of all roots is -1984. So, r1*r2*r3*r4 = -1984. Therefore, (-32)*(r3*r4) = -1984. Then, solving for r3*r4: (-32)*(r3*r4) = -1984 => r3*r4 = (-1984)/(-32) = 62. So r3*r4 = 62.So, now I know that the products of the roots in pairs: r1*r2 = -32 and r3*r4 = 62. So maybe we can group the roots into two quadratics? Let me think. If the quartic factors into (x² - (r1 + r2)x + r1r2)(x² - (r3 + r4)x + r3r4) = 0. Then expanding this should give the original quartic equation. Let's try that.Let me denote S = r1 + r2, and T = r3 + r4. Then the quartic factors as (x² - Sx -32)(x² - Tx +62) = 0. Let's multiply these two quadratics:First, multiply x² by x²: x⁴.Then, x² * (-Tx) = -T x³.x² * 62 = 62x².Then, -Sx * x² = -S x³.-Sx * (-Tx) = S T x².-Sx * 62 = -62 S x.-32 * x² = -32x².-32 * (-Tx) = 32 T x.-32 * 62 = -1984.Now, combining all the terms:x⁴ + (-T - S) x³ + (62 + S T - 32) x² + (-62 S + 32 T) x -1984 = 0.Comparing this to the original equation x⁴ - 18x³ + kx² + 200x -1984 = 0, we can equate coefficients:1. Coefficient of x³: -T - S = -18 => T + S = 18.2. Coefficient of x²: 62 + S T - 32 = k => S T + 30 = k.3. Coefficient of x: -62 S + 32 T = 200.So we have three equations here:1. S + T = 18.2. S T + 30 = k.3. -62 S + 32 T = 200.So we need to solve for S and T first. Let me write down the first equation: S + T = 18. So T = 18 - S. Substitute T into the third equation:-62 S + 32(18 - S) = 200.Compute 32*18: 32*10=320, 32*8=256, so 320+256=576. So:-62 S + 576 - 32 S = 200.Combine like terms: (-62 - 32) S + 576 = 200 => -94 S + 576 = 200.Subtract 576 from both sides: -94 S = 200 - 576 = -376.Therefore, S = (-376)/(-94) = 376/94. Let me compute that: 94*4 = 376, so 376/94 = 4. So S = 4. Then T = 18 - S = 14.So S = 4 and T = 14. Then, the coefficient k is S T + 30 = 4*14 + 30 = 56 + 30 = 86. So k = 86. Let me verify this.Check coefficients:1. S + T = 4 + 14 = 18. Correct.2. Coefficient of x²: 62 + (4)(14) -32 = 62 + 56 -32 = 118 -32 = 86. So k = 86. Correct.3. Coefficient of x: -62*4 + 32*14 = -248 + 448 = 200. Correct.So that works out. Therefore, the value of k is 86. Let me check if there's another way this could have been approached, maybe in case I made an assumption that the quartic factors into two quadratics with integer coefficients. Wait, but they told me the product of two roots is -32, which is an integer, so maybe the roots are integers? Let me check if the original quartic can be factored accordingly.If the quartic factors into (x² -4x -32)(x² -14x +62). Let me check if these quadratics multiply back to the original quartic. Let's compute (x² -4x -32)(x² -14x +62):First, x² * x² = x⁴.x²*(-14x) = -14x³.x²*62 = 62x².-4x *x² = -4x³.-4x*(-14x) = 56x².-4x*62 = -248x.-32*x² = -32x².-32*(-14x) = 448x.-32*62 = -1984.Now, combine all terms:x⁴ + (-14x³ -4x³) + (62x² +56x² -32x²) + (-248x +448x) -1984.Simplify each term:x⁴ -18x³ + (62+56-32)x² + (200x) -1984.Calculates:62+56=118; 118-32=86. So x⁴ -18x³ +86x² +200x -1984. Which matches the original equation. So yes, the factorization is correct. Therefore, the value of k is indeed 86. Therefore, the answer should be 86.Final AnswerThe value of ( k ) is boxed{86}.</think>Given the quartic equation (x^4 - 18x^3 + kx^2 + 200x - 1984 = 0), we know that the product of two of the roots is (-32). We need to determine the value of (k).Using Vieta's formulas, we know:1. The sum of the roots (r_1 + r_2 + r_3 + r_4 = 18).2. The product of all roots (r_1 r_2 r_3 r_4 = -1984).Given (r_1 r_2 = -32), we can find (r_3 r_4) using the product of all roots:[(-32)(r_3 r_4) = -1984 implies r_3 r_4 = frac{-1984}{-32} = 62]We assume the quartic equation factors into two quadratics ((x^2 - Sx - 32)(x^2 - Tx + 62)), where (S = r_1 + r_2) and (T = r_3 + r_4). Expanding this product and equating coefficients with the original equation, we get:1. Coefficient of (x^3): (-S - T = -18 implies S + T = 18)2. Coefficient of (x^2): (62 + ST - 32 = k implies ST + 30 = k)3. Coefficient of (x): (-62S + 32T = 200)Solving the system of equations:1. (S + T = 18)2. Substitute (T = 18 - S) into (-62S + 32T = 200):[-62S + 32(18 - S) = 200 implies -62S + 576 - 32S = 200 implies -94S + 576 = 200 implies -94S = -376 implies S = 4]3. Then (T = 18 - S = 14)Calculating (k):[k = ST + 30 = 4 times 14 + 30 = 56 + 30 = 86]Thus, the value of (k) is (boxed{86}).]

🔑:1. Let ( a, b, c, ) and ( d ) be the roots of the quartic equation ( x^4 - 18x^3 + kx^2 + 200x - 1984 = 0 ).2. By Vieta's formulas, we have the following relationships: [ a + b + c + d = 18 ] [ ab + ac + ad + bc + bd + cd = k ] [ abc + abd + acd + bcd = -200 ] [ abcd = -1984 ]3. We are given that the product of two of the roots is (-32). Without loss of generality, let ( ab = -32 ).4. Using the product of all roots: [ abcd = -1984 ] [ frac{abcd}{ab} = frac{-1984}{-32} = 62 ] Thus, ( cd = 62 ).5. We need to find ( k ). Using Vieta's formula for the sum of the products of the roots taken two at a time: [ ab + ac + ad + bc + bd + cd = k ] Substituting ( ab = -32 ) and ( cd = 62 ): [ k = -32 + ac + ad + bc + bd + 62 ] [ k = 30 + ac + ad + bc + bd ]6. We can factor ( ac + ad + bc + bd ) as ( (a + b)(c + d) ): [ ac + ad + bc + bd = (a + b)(c + d) ] Therefore: [ k = 30 + (a + b)(c + d) ]7. We know from Vieta's formulas that: [ a + b + c + d = 18 ] Let ( a + b = p ) and ( c + d = q ). Then: [ p + q = 18 ]8. We also know: [ ab = -32 quad text{and} quad cd = 62 ]9. Using the relationship for the sum of the products of the roots taken three at a time: [ abc + abd + acd + bcd = ab(c + d) + cd(a + b) = -200 ] Substituting ( ab = -32 ), ( cd = 62 ), and ( a + b = p ), ( c + d = q ): [ -32q + 62p = -200 ]10. Substituting ( q = 18 - p ): [ -32(18 - p) + 62p = -200 ] [ -576 + 32p + 62p = -200 ] [ 94p = 376 ] [ p = 4 ] Therefore: [ q = 18 - 4 = 14 ]11. Substituting ( p = 4 ) and ( q = 14 ) back into the equation for ( k ): [ k = 30 + (a + b)(c + d) ] [ k = 30 + 4 cdot 14 ] [ k = 30 + 56 ] [ k = 86 ]The final answer is ( boxed{86} ).

🧠:[<think>Okay, let's see. The problem is about Jackie and Phil each flipping three coins: two fair coins and a third coin with a 4/7 chance of heads. We need to find the probability that Jackie and Phil get the same number of heads, and then give the answer as m + n where the probability is m/n in simplest terms.First, I need to figure out the probability distribution for the number of heads when flipping these three coins. There are two fair coins and one biased coin. Let me break it down.Each fair coin has a 1/2 chance of heads and 1/2 tails. The third coin has a 4/7 chance of heads and 3/7 tails. Since the coins are independent, the number of heads can be 0, 1, 2, or 3. I need to compute the probability for each possible number of heads.Let's denote the two fair coins as Coin A and Coin B, and the biased coin as Coin C.For each number of heads k (0 to 3), I need to consider all possible combinations of the three coins that result in k heads. Then, compute the probability for each combination and sum them up.Let's start with k = 0 heads. That means all three coins must be tails.Probability of 0 heads:P(A=T, B=T, C=T) = (1/2)*(1/2)*(3/7) = (1/4)*(3/7) = 3/28.Next, k = 1 head. This can happen in three different ways:1. A is heads, B and C are tails.2. B is heads, A and C are tails.3. C is heads, A and B are tails.So, the probability is the sum of these three possibilities.Probability of 1 head:P(A=H, B=T, C=T) + P(A=T, B=H, C=T) + P(A=T, B=T, C=H)= (1/2)*(1/2)*(3/7) + (1/2)*(1/2)*(3/7) + (1/2)*(1/2)*(4/7)= (3/28) + (3/28) + (4/28)= (10/28) = 5/14.Wait, let me check the calculation again. Each of the first two terms is (1/2)(1/2)(3/7) = (1/4)(3/7) = 3/28, and the third term is (1/2)(1/2)(4/7) = (1/4)(4/7) = 1/7. So 3/28 + 3/28 + 1/7 = 3/28 + 3/28 + 4/28 = 10/28 = 5/14. That seems correct.For k = 2 heads. There are three ways again:1. A and B are heads, C is tails.2. A and C are heads, B is tails.3. B and C are heads, A is tails.Calculating each:First case: P(A=H, B=H, C=T) = (1/2)(1/2)(3/7) = 3/28.Second case: P(A=H, B=T, C=H) = (1/2)(1/2)(4/7) = 4/28 = 1/7.Third case: Similarly, P(A=T, B=H, C=H) = 1/7.So total probability for 2 heads:3/28 + 4/28 + 4/28 = 11/28.Wait, wait, 3/28 + 4/28 + 4/28 = 11/28? Let's check:First term: 3/28, second: 4/28, third: 4/28. So 3 + 4 + 4 = 11. Yes, 11/28.Then for k = 3 heads. All three coins must be heads.Probability of 3 heads:P(A=H, B=H, C=H) = (1/2)(1/2)(4/7) = (1/4)(4/7) = 1/7.So summarizing:k=0: 3/28k=1: 5/14 = 10/28k=2: 11/28k=3: 4/28 = 1/7Wait, 3 + 10 + 11 + 4 = 28? 3+10=13, 13+11=24, 24+4=28. Yes, that adds up. Good.So the probability distribution for the number of heads is:P(0) = 3/28P(1) = 10/28P(2) = 11/28P(3) = 4/28Simplify if needed. But maybe we can leave them as is for calculation purposes.Now, since both Jackie and Phil flip the same set of coins, their distributions are identical and independent. The probability that Jackie and Phil get the same number of heads is the sum over k=0 to 3 of [P(k)]^2.Because for each k, the probability that Jackie gets k heads and Phil gets k heads is [P(k)]^2, and since they are independent, we can sum these probabilities.So, the total probability is P(0)^2 + P(1)^2 + P(2)^2 + P(3)^2.Let me compute each term.First, P(0)^2 = (3/28)^2 = 9/784P(1)^2 = (10/28)^2 = 100/784P(2)^2 = (11/28)^2 = 121/784P(3)^2 = (4/28)^2 = 16/784Sum them up: 9 + 100 + 121 + 16 = 246. So total probability is 246/784.Simplify this fraction.First, check if 246 and 784 have a common factor.Divide numerator and denominator by 2: 246 ÷ 2 = 123; 784 ÷ 2 = 392. So 123/392.Check if 123 and 392 have a common factor. 123 factors into 3*41, and 392 is 49*8 = 7^2*2^3. So 123 is 3*41, 392 is 2^3*7^2. No common factors. So 123/392 is in simplest terms. Therefore, the probability is 123/392.Wait, but let me confirm the arithmetic again. 3^2 + 10^2 +11^2 +4^2 all over 28^2. 9 + 100 + 121 +16. 9+100=109, 109+121=230, 230+16=246. 246/784. Simplify: divide numerator and denominator by 2, get 123/392. 123 and 392. 123 divided by 3 is 41, 392 divided by 3 is 130.666... So no. 123 and 392 are co-prime? 123's prime factors: 3 and 41. 392 is 2^3*7^2. No overlap. So yes, 123/392 is simplified.Therefore, the answer is m + n = 123 + 392 = 515.Wait, but that seems high. Let me verify if my approach was correct.Alternative approach: The probability that two independent variables with the same distribution are equal is the sum over k of [P(k)]^2. That's a standard result. So if I correctly calculated P(k) for each k, then squaring and summing gives the right probability.But maybe I made a mistake in calculating P(k). Let me double-check.For k=0: All three tails. The two fair coins each have 1/2 chance of tails, and the biased coin has 3/7 chance of tails. So 1/2 * 1/2 * 3/7 = 3/28. Correct.k=1: Exactly one head. Three possibilities:1. Head on first fair coin, tails on the other two.Probability: 1/2 * 1/2 * 3/7 = 3/28.2. Head on second fair coin, tails on the other two.Same as above: 3/28.3. Head on biased coin, tails on both fair coins.Probability: 1/2 * 1/2 * 4/7 = 4/28.Total: 3 + 3 + 4 = 10/28. Correct.k=2: Exactly two heads. Three possibilities:1. Heads on both fair coins, tail on biased.Probability: 1/2 * 1/2 * 3/7 = 3. Maybe another verification. The integral of (3 -7x²) cos2x over 0 to 2π. ,<think>Okay, let me try to figure out this probability problem. So, we have Jackie and Phil each flipping three coins. Two of the coins are fair, and the third has a probability of 4/7 of landing heads. We need to find the probability that Jackie and Phil get the same number of heads. The answer should be in the form m/n where m and n are coprime, and then we have to find m + n. First, I need to understand the setup. Both Jackie and Phil are flipping three coins each: two fair coins and one biased coin with a 4/7 chance of heads. They each do this once, and we want the probability that the number of heads Jackie gets is equal to the number Phil gets. To solve this, I think I need to compute the probability distribution for the number of heads each person gets, and then find the probability that both Jackie and Phil have the same number of heads. Since both are flipping the same set of coins, their distributions should be identical, right? So maybe I can model this as a single probability distribution for the number of heads in three flips (two fair, one biased), and then the probability that two independent trials of this distribution result in the same value is the sum over all k of [P(k)]², where P(k) is the probability of getting k heads. Let me verify that. If two independent variables X and Y have the same distribution, then P(X = Y) is indeed the sum over k of [P(X = k)]². So that seems correct. So, my approach should be:1. Find the probability distribution for the number of heads when flipping two fair coins and one biased coin.2. For each possible number of heads (0, 1, 2, 3), square the probability and sum them up.3. That sum will be the probability that Jackie and Phil have the same number of heads.4. Then, simplify the fraction and add the numerator and denominator.Okay, let's start by finding the probability distribution for the number of heads. The coins are two fair coins and one biased coin with P(heads) = 4/7. Let's denote the number of heads as k, which can be 0, 1, 2, or 3.To compute P(k), we need to consider all possible ways to get k heads from the three coins. The two fair coins each have a probability of 1/2 for heads and 1/2 for tails, and the biased coin has 4/7 for heads and 3/7 for tails.Since the coins are independent, the total probability for a specific combination is the product of the probabilities of each individual outcome.Let's compute P(0), P(1), P(2), P(3).Starting with P(0): the probability that all three coins are tails. For the two fair coins, the probability of tails is 1/2 each, and for the biased coin, it's 3/7. So:P(0) = (1/2) * (1/2) * (3/7) = (1/4) * (3/7) = 3/28.Next, P(1): the probability of exactly one head. There are three possibilities here: the head could be from the first fair coin, the second fair coin, or the biased coin. Let's compute each:1. Head on the first fair coin, tails on the second and the biased coin:(1/2) * (1/2) * (3/7) = 3/28.2. Head on the second fair coin, tails on the first and the biased coin:(1/2) * (1/2) * (3/7) = 3/28.3. Head on the biased coin, tails on both fair coins:(1/2) * (1/2) * (4/7) = (1/4) * (4/7) = 1/7.Adding these up: 3/28 + 3/28 + 1/7 = (3 + 3 + 4)/28 = 10/28 = 5/14.So P(1) = 5/14.Moving on to P(2): exactly two heads. Again, three possibilities: heads on both fair coins and tail on the biased, heads on the first fair and the biased coin, or heads on the second fair and the biased coin.1. Both fair coins are heads, biased coin is tail:(1/2 * 1/2) * 3/7 = (1/4) * 3/7 = 3/28.2. First fair coin head, second fair tail, biased head:(1/2 * 1/2) * 4/7 = (1/4) * 4/7 = 1/7.3. Second fair coin head, first fair tail, biased head:Same as above: 1/7.Adding these up: 3/28 + 1/7 + 1/7 = 3/28 + 2/7 = 3/28 + 8/28 = 11/28.Therefore, P(2) = 11/28.Lastly, P(3): all three coins are heads. This requires both fair coins and the biased coin to be heads:(1/2 * 1/2) * 4/7 = (1/4) * 4/7 = 1/7.So P(3) = 1/7.Let me check if the probabilities sum to 1:P(0) + P(1) + P(2) + P(3) = 3/28 + 5/14 + 11/28 + 1/7Convert all to 28 denominators:3/28 + 10/28 + 11/28 + 4/28 = (3 + 10 + 11 + 4)/28 = 28/28 = 1. Okay, that checks out.So the probabilities are:P(0) = 3/28,P(1) = 5/14,P(2) = 11/28,P(3) = 1/7.Now, the next step is to compute the sum of the squares of these probabilities. That is, P(Jackie = Phil) = P(0)^2 + P(1)^2 + P(2)^2 + P(3)^2.Let's compute each term:P(0)^2 = (3/28)^2 = 9/784,P(1)^2 = (5/14)^2 = 25/196,P(2)^2 = (11/28)^2 = 121/784,P(3)^2 = (1/7)^2 = 1/49.Now, sum these up:9/784 + 25/196 + 121/784 + 1/49.First, let's convert all fractions to have a denominator of 784.9/784 remains as is.25/196: Since 196 * 4 = 784, multiply numerator and denominator by 4: 25*4=100, so 100/784.121/784 remains as is.1/49: 49 * 16 = 784, so multiply numerator and denominator by 16: 1*16=16, so 16/784.Therefore, adding them together:9 + 100 + 121 + 16 = let's compute step by step.9 + 100 = 109,109 + 121 = 230,230 + 16 = 246.So total is 246/784.Now, simplify this fraction. Let's see if 246 and 784 have a common divisor.First, divide numerator and denominator by 2: 246/2 = 123, 784/2 = 392.123/392. Check if they have a common divisor. 123 is 3*41, 392 is 16*24.5? Wait, 392 divided by 2 is 196, divided by 2 again is 98, again 49, which is 7^2. So 392 = 2^3 * 7^2. 123 is 3*41. 41 is prime. So 123 and 392 have no common factors except 1. Therefore, 123/392 is the simplified form.Wait, 123 is 3*41, 392 is 2^3*7^2. No common factors, so yes, 123/392. But wait, the problem states that m/n is the probability where m and n are coprime. So 123 and 392. Let me check GCD(123, 392).Divisors of 123: 1, 3, 41, 123.Divisors of 392: 1, 2, 4, 7, 8, 14, 28, 49, 56, 98, 196, 392. The only common divisor is 1. So yes, 123/392 is reduced. But wait, 123 + 392 is 515. But I have a feeling that maybe I made a miscalculation here. Let me check the sum again.Wait, 9/784 + 25/196 + 121/784 + 1/49. Let me verify each conversion:25/196 to 100/784: Correct, since 25*4=100 and 196*4=784.1/49 to 16/784: Correct, since 1*16=16 and 49*16=784.9 + 100 + 121 +16: 9+100=109, 109+121=230, 230+16=246. So 246/784, simplifies to 123/392. So m=123, n=392. Are they coprime? 123 factors into 3×41, 392 factors into 2³×7². No common factors. So yes, 123/392. Then m + n = 123 + 392 = 515.But wait, the answer seems a bit high. I wonder if I made a mistake here. Let me double-check each step.First, verifying the probabilities:P(0) = 3/28: Correct. Two tails (1/2 each) and one tail on the biased (3/7): (1/2)^2 * 3/7 = 1/4 * 3/7 = 3/28.P(1): We had three cases. First two fair coins each contribute 3/28, and the biased case is 1/7. So 3/28 + 3/28 + 1/7 = 6/28 + 4/28 = 10/28 = 5/14. Correct.P(2): Similarly, three cases. Two fair heads and biased tail: 3/28. Each fair plus biased: 1/7 each. So 3/28 + 1/7 +1/7 = 3/28 + 8/28 = 11/28. Correct.P(3): Both fair and biased: (1/2)^2 *4/7 = 1/4 *4/7=1/7. Correct.Then, squaring each probability:(3/28)^2=9/784,(5/14)^2=25/196,(11/28)^2=121/784,(1/7)^2=1/49.Convert to 784 denominator:25/196 = 100/784,1/49=16/784.So total sum: 9 + 100 + 121 +16=246. 246/784=123/392. Yes, that seems right.But 515 seems like a big number. Wait, is the answer 123 + 392 = 515? Let me check if 123/392 is indeed in simplest terms. 123 divided by 3 is 41, 392 divided by 3 is 130.666… So no. Divided by 41: 392 divided by 41 is 9.56… Not an integer. So yeah, they are coprime. So the answer is 515.But wait, hold on. Let me check again. Maybe I made a miscalculation in adding up the squares. Let me recalculate the sum:9 (from 0 heads) + 100 (from 1 head) + 121 (from 2 heads) +16 (from 3 heads) = 246. Then 246 divided by 784.Wait, 246 divided by 784. Let me check if that's 123/392. Yes, because 246 divided by 2 is 123, 784 divided by 2 is 392. So correct.But 123 and 392: 123 is 3*41, 392 is 2^3 *7^2. No common factors. So 123/392 reduces to 123/392. So m=123, n=392, sum is 515.But wait, I recall that in some similar problems, sometimes the answer is smaller. Let me check if the approach is correct. The idea is that the probability that Jackie and Phil have the same number of heads is equal to the sum over k of [P(k)]². Is that correct?Yes, if X and Y are independent and identically distributed random variables, then P(X = Y) = Σ_{k} P(X = k) * P(Y = k) = Σ_{k} [P(k)]². So that is correct.Therefore, the steps seem correct. So unless there's a miscalculation in P(k) values, which I checked, then the answer should be 515. Hmm. But 515 seems like an unusual answer. Let me check again.Alternatively, maybe there's another approach. Let's see: compute the probability that Jackie and Phil both get 0 heads: (3/28)^2. Both get 1 head: (5/14)^2. Both get 2 heads: (11/28)^2. Both get 3 heads: (1/7)^2. Then sum these up. So same as above.Alternatively, maybe the question is considering the biased coin as the third coin, but maybe the two fair coins are distinguishable? Wait, no, the coins are just coins. The count is based on the number of heads, not which specific coins. So the probabilities are correctly calculated. Alternatively, maybe I miscalculated the probabilities. Let me check P(2) again. For two heads, there are three cases:1. Both fair coins are heads, biased is tail: (1/2)^2*(3/7) = 1/4 *3/7=3/28.2. First fair head, biased head: (1/2)(1/2)(4/7)=1/4*4/7=1/7.3. Second fair head, biased head: same as above, 1/7.Total P(2)=3/28 +1/7 +1/7=3/28 +2/7=3/28 +8/28=11/28. Correct.Similarly, P(1)=5/14, correct.So, everything seems okay. So the answer is 515. Maybe that's the correct answer.Wait, but just to be thorough, let me compute it again step by step.Original sum: [3² + (5/14 *28)² +11² + (1/7*28)^2] over 28²? Wait, no. Wait, the probabilities were:P(0) =3/28,P(1)=5/14=10/28,P(2)=11/28,P(3)=4/28.Wait, wait, P(3)=1/7=4/28. Wait, 1/7 is 4/28? 1/7 is 4/28? No, 1/7 is 4/28? Wait 1/7 is approximately 0.1428, 4/28 is 1/7 ≈0.1428. Yes, so 1/7=4/28. But in our previous calculation, P(3)=1/7=4/28. Wait, but when we calculated P(3), we had (1/2)^2*(4/7)=1/4*4/7=1/7=4/28. Correct.But in our sum of squares, we had:(3/28)^2 + (10/28)^2 + (11/28)^2 + (4/28)^2 = [9 + 100 + 121 +16]/784 =246/784=123/392.Yes, because all terms were converted to /28 first. So, that's correct. Therefore, the answer is 123/392, sum is 515. Therefore, unless I made a mistake in calculating the original P(k)s, which I don't see, then 515 should be the answer.But just to confirm, let's compute the sum of squares again, using fractions:Sum = (3/28)^2 + (5/14)^2 + (11/28)^2 + (1/7)^2Compute each term:3/28 squared: 9/784.5/14 squared: 25/196. 25/196 is equal to (25*4)/784 = 100/784.11/28 squared: 121/784.1/7 squared: 1/49. 1/49 is equal to 16/784.Therefore, total sum is 9 + 100 + 121 +16 =246 over 784. 246 divided by 784. 246 divided by 2 is 123, 784 divided by 2 is 392. 123/392. Correct.Yes, so 515 is the answer. I think that's correct.Final Answerboxed{515}</think>Jackie and Phil each flip three coins: two fair coins and one biased coin with a probability of (frac{4}{7}) of landing heads. We need to find the probability that Jackie and Phil get the same number of heads.First, we compute the probability distribution for the number of heads (0, 1, 2, or 3) when flipping the three coins. The probabilities are calculated as follows:- Probability of 0 heads (P(0)): [ P(0) = left(frac{1}{2}right)^2 cdot frac{3}{7} = frac{1}{4} cdot frac{3}{7} = frac{3}{28} ]- Probability of 1 head (P(1)): [ P(1) = 2 left(frac{1}{2} cdot frac{1}{2} cdot frac{3}{7}right) + left(frac{1}{2} cdot frac{1}{2} cdot frac{4}{7}right) = 2 cdot frac{3}{28} + frac{1}{7} = frac{6}{28} + frac{4}{28} = frac{10}{28} = frac{5}{14} ]- Probability of 2 heads (P(2)): [ P(2) = left(frac{1}{2}right)^2 cdot frac{3}{7} + 2 left(frac{1}{2} cdot frac{1}{2} cdot frac{4}{7}right) = frac{3}{28} + 2 cdot frac{4}{28} = frac{3}{28} + frac{8}{28} = frac{11}{28} ]- Probability of 3 heads (P(3)): [ P(3) = left(frac{1}{2}right)^2 cdot frac{4}{7} = frac{1}{4} cdot frac{4}{7} = frac{1}{7} ]Next, we calculate the probability that Jackie and Phil get the same number of heads by summing the squares of these probabilities:- Squared probabilities: [ begin{aligned} P(0)^2 &= left(frac{3}{28}right)^2 = frac{9}{784}, P(1)^2 &= left(frac{5}{14}right)^2 = frac{25}{196} = frac{100}{784}, P(2)^2 &= left(frac{11}{28}right)^2 = frac{121}{784}, P(3)^2 &= left(frac{1}{7}right)^2 = frac{1}{49} = frac{16}{784}. end{aligned} ]- Sum of squared probabilities: [ frac{9}{784} + frac{100}{784} + frac{121}{784} + frac{16}{784} = frac{246}{784} = frac{123}{392} ]The fraction (frac{123}{392}) is in its simplest form, so (m = 123) and (n = 392). Therefore, (m + n = 123 + 392 = 515).[boxed{515}]]

🔑:1. Determine the probability distribution for the number of heads in one flip of the three coins: - Let ( X ) be the number of heads in one flip of the three coins. - The two fair coins each have a probability of ( frac{1}{2} ) for heads, and the biased coin has a probability of ( frac{4}{7} ) for heads. - The possible values for ( X ) are 0, 1, 2, and 3 heads.2. Calculate the probability for each possible number of heads: - 0 heads: [ P(X = 0) = left( frac{1}{2} right)^2 cdot left( frac{3}{7} right) = frac{1}{4} cdot frac{3}{7} = frac{3}{28} ] - 1 head: - One fair coin shows heads, the other fair coin shows tails, and the biased coin shows tails: [ 2 cdot left( frac{1}{2} right)^2 cdot left( frac{3}{7} right) = 2 cdot frac{1}{4} cdot frac{3}{7} = frac{6}{28} ] - The biased coin shows heads, and both fair coins show tails: [ left( frac{1}{2} right)^2 cdot left( frac{4}{7} right) = frac{1}{4} cdot frac{4}{7} = frac{4}{28} ] - Total probability for 1 head: [ P(X = 1) = frac{6}{28} + frac{4}{28} = frac{10}{28} ] - 2 heads: - Two fair coins show heads, and the biased coin shows tails: [ left( frac{1}{2} right)^2 cdot left( frac{3}{7} right) = frac{1}{4} cdot frac{3}{7} = frac{3}{28} ] - One fair coin shows heads, the other fair coin shows tails, and the biased coin shows heads: [ 2 cdot left( frac{1}{2} right)^2 cdot left( frac{4}{7} right) = 2 cdot frac{1}{4} cdot frac{4}{7} = frac{8}{28} ] - Total probability for 2 heads: [ P(X = 2) = frac{3}{28} + frac{8}{28} = frac{11}{28} ] - 3 heads: [ P(X = 3) = left( frac{1}{2} right)^2 cdot left( frac{4}{7} right) = frac{1}{4} cdot frac{4}{7} = frac{4}{28} ]3. Calculate the probability that Jackie and Phil get the same number of heads: - The probability that both get 0 heads: [ left( frac{3}{28} right)^2 = frac{9}{784} ] - The probability that both get 1 head: [ left( frac{10}{28} right)^2 = frac{100}{784} ] - The probability that both get 2 heads: [ left( frac{11}{28} right)^2 = frac{121}{784} ] - The probability that both get 3 heads: [ left( frac{4}{28} right)^2 = frac{16}{784} ]4. Sum the probabilities: [ frac{9}{784} + frac{100}{784} + frac{121}{784} + frac{16}{784} = frac{246}{784} ]5. Simplify the fraction: [ frac{246}{784} = frac{123}{392} ]6. Find ( m + n ) where ( frac{m}{n} = frac{123}{392} ): [ m = 123, quad n = 392 quad Rightarrow quad m + n = 123 + 392 = 515 ]The final answer is ( boxed{515} )

🧠:[<think>Okay, so I need to find the sum of the first 2016 terms of the sequence where each term is a_n = 2^n * cos(nπ/2). Hmm, let me start by understanding this sequence better. First, let's recall that cos(nπ/2) has some periodicity. The cosine function has a period of 2π, but when the angle is multiplied by π/2, the period might change. Let me compute the first few terms to see if there's a pattern.For n = 1: cos(π/2) = 0, so a_1 = 2^1 * 0 = 0For n = 2: cos(π) = -1, so a_2 = 2^2 * (-1) = -4For n = 3: cos(3π/2) = 0, so a_3 = 2^3 * 0 = 0For n = 4: cos(2π) = 1, so a_4 = 2^4 * 1 = 16For n = 5: cos(5π/2) = 0, so a_5 = 2^5 * 0 = 0For n = 6: cos(3π) = -1, so a_6 = 2^6 * (-1) = -64For n = 7: cos(7π/2) = 0, so a_7 = 2^7 * 0 = 0For n = 8: cos(4π) = 1, so a_8 = 2^8 * 1 = 256Wait a second, so the sequence alternates between 0, -4, 0, 16, 0, -64, 0, 256, and so on. It looks like every odd term is zero, and the even terms alternate between negative and positive, increasing in magnitude by a factor of 4 each time. Let me verify this.If n is even, let's let n = 2k where k is a positive integer. Then cos(nπ/2) = cos(kπ) = (-1)^k. So a_{2k} = 2^{2k} * (-1)^k. That is, (2^2)^k * (-1)^k = 4^k * (-1)^k = (-4)^k. So the even terms are -4, 16, -64, 256, ..., which is indeed a geometric sequence with first term -4 and common ratio -4.But wait, for k=1, a_2 = (-4)^1 = -4, which is correct. For k=2, a_4 = (-4)^2 = 16, correct. k=3, a_6 = (-4)^3 = -64, yes. So the even terms form a geometric sequence with ratio r = -4, starting at n=2 (term index k=1). Therefore, the sum S_n of the first n terms is equivalent to summing the even terms up to the floor(n/2) terms. But since every odd term is zero, S_{2016} is just the sum of the first 2016 terms, which includes 1008 even terms (since every other term is non-zero). Wait, 2016 divided by 2 is 1008, so there are 1008 non-zero terms. So S_{2016} is equal to the sum of the first 1008 terms of the geometric sequence starting with -4, ratio -4.Therefore, the sum would be S = a_1*(r^{m} - 1)/(r - 1), where a_1 is the first term, r is the common ratio, and m is the number of terms. Here, a_1 = -4, r = -4, m = 1008.Plugging into the formula: S = (-4)*[(-4)^{1008} - 1]/[(-4) - 1] = (-4)*[(-4)^{1008} - 1]/(-5) = (-4)/-5 * [(-4)^{1008} - 1] = (4/5)*[(-4)^{1008} - 1].But let's simplify (-4)^{1008}. Since the exponent is even, this is equal to (4)^{1008} because (-4)^{1008} = (4)^{1008} as the negative sign is raised to an even power. So, (-4)^{1008} = 4^{1008}. Therefore, the expression becomes (4/5)*(4^{1008} - 1).Hmm, so S_{2016} = (4/5)*(4^{1008} - 1). But maybe we can write this in a different way. Note that 4^{1008} is (2^2)^{1008} = 2^{2016}. So 4^{1008} = 2^{2016}. Therefore, substituting back, we have S_{2016} = (4/5)*(2^{2016} - 1).Wait, but let me check the steps again to make sure I didn't make a mistake. So, the key points: every even term is (-4)^k where k is the term number in the even sequence (k=1 corresponds to n=2, k=2 to n=4, etc.), and there are 1008 such terms. The sum of a geometric series is a1*(r^m -1)/(r -1). So:First term a1 = -4, ratio r = -4, number of terms m=1008. Then:Sum = (-4)*[(-4)^{1008} - 1]/[(-4) - 1] = (-4)*[4^{1008} - 1]/(-5) because (-4)^{1008}=4^{1008}. Then, the numerator becomes (-4)*(4^{1008} - 1) and denominator is -5. So, (-4)/(-5) is 4/5, so total sum is (4/5)*(4^{1008} -1). Which is the same as (4/5)*(2^{2016} -1). But 4^{1008} is (2^2)^{1008} = 2^{2016}, yes. So both expressions are equivalent.Alternatively, perhaps the problem expects the answer in terms of 4^{1008} or 2^{2016}. Let's see which is simpler. Both are correct, but maybe 4^{1008} is more straightforward given the original ratio was -4. However, 2^{2016} might be considered simpler. But in any case, the answer is (4/5)*(4^{1008} -1) or (4/5)*(2^{2016} -1). Let me check if that's the correct expression.Wait, but maybe there's a different approach. Let me think. Alternatively, perhaps using complex numbers. Since cos(nπ/2) is the real part of e^{i n π/2}, but maybe that complicates things. Alternatively, recursion?Alternatively, since the non-zero terms are every even term, forming a geometric series with ratio -4, and 1008 terms. Then the sum is as calculated. So unless there's a telescoping sum or another way to simplify, this seems like the correct approach.But let me confirm with a smaller number. Let's compute S_4. According to the formula, S_4 should be the sum of first 4 terms: 0 + (-4) + 0 +16 = 12. Using the formula, m = 4/2 = 2 terms. Then sum is (-4)*( (-4)^2 -1 )/( -4 -1 ) = (-4)*(16 -1)/(-5) = (-4)*(15)/(-5) = (-4)*(-3) = 12. Correct. Similarly, S_6 would be 0 + (-4) +0 +16 +0 + (-64) = (-4) +16 + (-64) = (-52). Using the formula, m=3 terms. Sum = (-4)*( (-4)^3 -1 )/( -4 -1 ) = (-4)*( -64 -1 )/(-5) = (-4)*(-65)/(-5) = (-4)*(13) = -52. Correct.So the formula works for smaller n, so likely correct. Therefore, S_{2016} is (4/5)*(4^{1008} -1). Alternatively, since 4^{1008} is 2^{2016}, so S_{2016} = (4/5)(2^{2016} -1). Either form is acceptable, but maybe the problem expects the answer in terms of 2^{2016} because the original formula uses 2^n. So perhaps that's preferable.But let's check if 4^{1008} is equal to 2^{2016}. Since 4 is 2 squared, so 4^{1008} = (2^2)^{1008} = 2^{2*1008} = 2^{2016}. Yes. So both forms are equivalent.Therefore, S_{2016} = (4/5)(2^{2016} -1). Let me just write that as boxed{dfrac{4(2^{2016} - 1)}{5}}.But let me verify once again. Let me compute S_8. The actual sum would be 0 + (-4) + 0 +16 +0 + (-64) +0 +256. The sum is (-4) +16 + (-64) +256 = (12) + (192) = 204. Using the formula, m=4 terms. Sum = (-4)*( (-4)^4 -1 )/( -4 -1 ) = (-4)*(256 -1)/(-5) = (-4)*(255)/(-5) = (-4)*(-51) = 204. Correct. So the formula works.Therefore, confident that S_{2016} is (4/5)(2^{2016} -1). The answer is likely this. Therefore, boxed as boxed{dfrac{4}{5} left(2^{2016} - 1right)}.Final Answerboxed{dfrac{4}{5} left(2^{2016} - 1right)}</think>The general term formula of the sequence ({a_n}) is given by (a_n = 2^n cos frac{npi}{2}). To find (S_{2016}), the sum of the first 2016 terms, we observe the periodicity of (cos frac{npi}{2}):- For odd (n), (cos frac{npi}{2} = 0), so the corresponding term (a_n) is 0.- For even (n = 2k), (cos frac{2kpi}{2} = cos(kpi) = (-1)^k), so the corresponding term (a_{2k} = 2^{2k} (-1)^k = (-4)^k).Thus, the non-zero terms form a geometric sequence with the first term (-4) and common ratio (-4). Since there are 2016 terms in total, there are 1008 non-zero terms. The sum of the first 1008 terms of this geometric sequence is calculated as follows:1. The first term (a_1 = -4).2. The common ratio (r = -4).3. The number of terms (m = 1008).The sum (S) of the first (m) terms of a geometric series is given by:[S = a_1 frac{r^m - 1}{r - 1}]Substituting the values:[S = -4 frac{(-4)^{1008} - 1}{-4 - 1} = -4 frac{4^{1008} - 1}{-5} = frac{4}{5} (4^{1008} - 1)]Since (4^{1008} = (2^2)^{1008} = 2^{2016}), we can rewrite the sum as:[S = frac{4}{5} (2^{2016} - 1)]Thus, the value of (S_{2016}) is:[boxed{dfrac{4}{5} left(2^{2016} - 1right)}],<think>Okay, so I need to find the sum of the first 2016 terms of the sequence where each term is given by a_n = 2^n * cos(nπ/2). Hmm, let's start by understanding this sequence better. First, cosine of nπ/2. Let me think about the values of cos(nπ/2) for different n. The cosine function here has a period, right? Since cos(θ) has a period of 2π, but here the angle is increasing by π/2 each time. Let's see:When n=1: cos(π/2) = 0n=2: cos(π) = -1n=3: cos(3π/2) = 0n=4: cos(2π) = 1n=5: cos(5π/2) = 0n=6: cos(3π) = -1n=7: cos(7π/2) = 0n=8: cos(4π) = 1So it seems like every odd term is zero. The even terms alternate between -1 and 1. Wait, let's check again:Wait n=2: cos(π) is indeed -1n=4: cos(2π) is 1n=6: cos(3π) is -1n=8: cos(4π) is 1, etc.So actually, the terms where n is even will alternate between -1 and 1, multiplied by 2^n, and the odd terms are all zero. So the sequence is 0, -2^2, 0, 2^4, 0, -2^6, 0, 2^8, and so on.Therefore, the sequence {a_n} is: 0, -4, 0, 16, 0, -64, 0, 256, 0, -1024, ... and so on.Therefore, when we take the sum S_n, since every odd term is zero, only the even terms contribute. Moreover, the even terms can be indexed as k = 1, 2, 3, ... with n = 2k. So for each k from 1 to 1008 (since 2016 is the total number of terms, so the even terms would be up to k=2016/2=1008).Wait, but let me confirm: If the original sequence has 2016 terms, then the number of even terms is 1008, since every other term is even. So S_2016 is the sum of the first 2016 terms, which includes 1008 even terms and 1008 odd terms, all of which are zero. Therefore, the sum S_2016 is equal to the sum of the first 1008 even terms.So, reindexing the even terms. Let's let k = 1 to 1008, corresponding to n = 2k. Then the term a_{2k} = 2^{2k} * cos(2kπ/2) = 2^{2k} * cos(kπ). But cos(kπ) is (-1)^k. Therefore, a_{2k} = 2^{2k} * (-1)^k. So each even term is (4^k)*(-1)^k = (-4)^k.Therefore, the sum S_2016 is the sum from k=1 to k=1008 of (-4)^k.Wait, that would be a geometric series with first term (-4)^1 = -4, common ratio r = -4, and number of terms 1008. So the sum is Sum_{k=1}^{1008} (-4)^k.But wait, the formula for the sum of a geometric series is S = a1*(r^n - 1)/(r - 1) when r ≠ 1. But here, the first term is a1 = -4, ratio r = -4, and number of terms is 1008.Therefore, the sum should be:Sum = (-4) * [ (-4)^{1008} - 1 ] / [ (-4) - 1 ]Simplify denominator: (-4 - 1) = -5Numerator: (-4)^{1008} - 1But note that (-4)^{1008} = ( (-4)^2 )^{504} = (16)^{504} = 4^{1008}Wait, but (-4)^{1008} is positive because exponent is even. So (-4)^{1008} = 4^{1008}Therefore, numerator becomes 4^{1008} - 1Thus, Sum = (-4)*(4^{1008} - 1)/(-5) = (-4)*(4^{1008} - 1)/(-5)Multiply numerator and denominator:(-4)/(-5) = 4/5So Sum = (4/5)*(4^{1008} - 1)But 4^{1008} = (2^2)^{1008} = 2^{2016}Therefore, Sum = (4/5)*(2^{2016} - 1)Wait, but let me check the steps again.Original sum: Sum_{k=1}^{1008} (-4)^kThis is a geometric series with first term a = -4, ratio r = -4, n = 1008 terms.Sum = a*(r^n - 1)/(r - 1) = (-4)*[ (-4)^1008 - 1 ] / [ (-4) - 1 ] = (-4)*(4^{1008} - 1)/(-5) = [ -4*(4^{1008} - 1) ] / (-5) = 4*(4^{1008} - 1)/5 = (4^{1009} - 4)/5Wait, but 4*(4^{1008}) is 4^{1009}So Sum = (4^{1009} - 4)/5Alternatively, expressed as (4/5)(4^{1008} - 1)But both are equivalent.Alternatively, 4^{1009} = 2^{2018}But maybe better to leave it as 4^{1009}But let me check again with a small example to make sure.Suppose we have Sum_{k=1}^m (-4)^kLet m=1: Sum = -4. Formula: ( -4*( (-4)^1 -1 ) ) / ( -4 -1 ) = (-4*(-4 -1))/(-5) = (-4*(-5))/(-5)= 20/-5 = -4. Correct.m=2: Sum = -4 + 16 = 12. Formula: (-4*( (-4)^2 -1 ))/(-5) = (-4*(16 -1))/(-5) = (-4*15)/(-5) = (-60)/(-5)=12. Correct.m=3: Sum = -4 +16 -64 = -52. Formula: (-4*( (-4)^3 -1 ))/(-5) = (-4*(-64 -1))/(-5) = (-4*(-65))/(-5) = (260)/(-5) = -52. Correct.So the formula works. Therefore, the sum is indeed (4^{1009} -4)/5.But the question says to find S_{2016}, which is this sum. Therefore, the answer is (4^{1009} -4)/5.But let me check again if I made a mistake in the indexing.Wait, when n=2k, for k=1 to 1008, that gives the even terms from n=2 to n=2016. So 1008 terms. Therefore, Sum_{k=1}^{1008} a_{2k} = Sum_{k=1}^{1008} (-4)^k. So that's correct.But when n=1, original term is a_1=0, so the sum starts contributing from a_2=-4, which is k=1.Therefore, the sum is indeed Sum_{k=1}^{1008} (-4)^k.So the sum is ( -4*( (-4)^{1008} -1 ) ) / ( -4 -1 ) = (-4*(4^{1008} -1 ))/(-5) = (4*(4^{1008}-1))/5 = (4^{1009} -4)/5. Yes, that's correct.Therefore, the answer is (4^{1009} -4)/5.But perhaps the problem expects a simplified form, maybe in terms of 2^{something}?Wait, 4^{1009} is (2^2)^{1009} = 2^{2018}, and 4 is 2^2. So we can write:(2^{2018} - 2^2)/5 = (2^2*(2^{2016} -1))/5 = 4*(2^{2016} -1)/5Alternatively, 4^{1009} -4 = 4*(4^{1008} -1) = 4*( (4)^{1008} -1 )So both forms are possible, but perhaps the former is better.But the question says "put your final answer within boxed{}". So maybe write in the simplest form. Let's check.Expressed as (4^{1009} -4)/5 is straightforward. Alternatively, 4*(4^{1008} -1)/5. Either is acceptable, but perhaps the first is better. However, let's check if this can be simplified further. Since 4^{1009} = 4*4^{1008}, so (4*4^{1008} -4)/5 = 4(4^{1008} -1)/5. Which is equivalent. So either way.Alternatively, using 2^{2018} -4 over 5. But perhaps they prefer one form over another.Alternatively, maybe the problem can be approached differently. Let's think again.The sequence a_n = 2^n cos(nπ/2). So, this is equivalent to 2^n multiplied by cos(nπ/2). As we saw, the odd terms are zero, and even terms alternate between -4^k and 4^k where k is the index of the even term.Alternatively, perhaps using complex numbers. Let me think.Since cos(nπ/2) can be related to the real part of e^{i n π/2}, which is i^n. Because e^{i θ} = cosθ + i sinθ, so e^{i nπ/2} = cos(nπ/2) + i sin(nπ/2). Therefore, cos(nπ/2) is the real part of i^n. But i^n cycles through 1, i, -1, -i, etc. Therefore, the real part is 1, 0, -1, 0, 1, 0, -1, 0,... which matches our previous conclusion.Therefore, a_n = 2^n * Re(i^n) = Re( (2i)^n )Therefore, the sum S_n = Sum_{k=1}^n Re( (2i)^k ) = Re( Sum_{k=1}^n (2i)^k )Therefore, the sum S_n is the real part of the geometric series Sum_{k=1}^n (2i)^k.The sum of a geometric series is a*(r^{n} -1)/(r -1), where a is the first term, r is the ratio. Here, a=2i, r=2i.Therefore, Sum_{k=1}^n (2i)^k = 2i*( (2i)^n -1 ) / (2i -1 )Then S_n is the real part of this.But in our case, n=2016. Therefore, S_{2016} = Re[ 2i*( (2i)^{2016} -1 ) / (2i -1 ) ]Let me compute this.First, compute (2i)^{2016}. Since 2i is 2*e^{iπ/2}, so (2i)^2016 = 2^{2016} * e^{iπ/2 *2016} = 2^{2016} * e^{i1008π} = 2^{2016} * (cos1008π + i sin1008π). Since 1008 is even, e^{i1008π} = e^{i2π*504} = 1. Therefore, (2i)^{2016} = 2^{2016}*1 = 2^{2016}Therefore, (2i)^{2016} -1 = 2^{2016} -1So, Sum_{k=1}^{2016} (2i)^k = 2i*(2^{2016} -1 )/(2i -1 )Therefore, S_{2016} = Re[ 2i*(2^{2016} -1 )/(2i -1 ) ]Now, compute the denominator: 2i -1 = -1 + 2iMultiply numerator and denominator by the conjugate of the denominator to make it real.The conjugate of -1 + 2i is -1 -2i.Therefore:[2i*(2^{2016} -1 )] * (-1 -2i) / [ (-1 +2i)(-1 -2i) ]First compute the denominator:(-1)^2 - (2i)^2 = 1 -4i^2 = 1 -4*(-1) = 1 +4=5Numerator:2i*(2^{2016} -1 )*(-1 -2i) = 2i*(-1 -2i)*(2^{2016} -1 )First compute 2i*(-1 -2i):= -2i -4i^2 = -2i -4(-1) = -2i +4 = 4 -2iThen multiply by (2^{2016} -1 ):= (4 -2i)*(2^{2016} -1 )Therefore, numerator is (4 -2i)*(2^{2016} -1 )Therefore, the entire expression is [ (4 -2i)*(2^{2016} -1 ) ] /5Therefore, S_{2016} = Re[ (4 -2i)*(2^{2016} -1 ) /5 ] = Re[ (4*(2^{2016}-1 ) -2i*(2^{2016}-1 )) /5 ]Taking the real part:= [4*(2^{2016}-1 )]/5 - Re[2i*(2^{2016}-1 )/5]But the second term has a factor of i, so its real part is zero.Therefore, S_{2016} = [4*(2^{2016} -1 )]/5Which is the same as (4*2^{2016} -4)/5 = (2^{2018} -4)/5Wait, but earlier we had (4^{1009} -4)/5. But 4^{1009} = (2^2)^{1009} = 2^{2018}, so yes, this is the same result.Therefore, both methods confirm that S_{2016} = (2^{2018} -4)/5Alternatively, 4*(2^{2016} -1)/5.But 2^{2018} is 4*2^{2016}, so (4*2^{2016} -4)/5 = 4*(2^{2016} -1)/5So both forms are equivalent.Which one is simpler? Maybe (2^{2018} -4)/5 is more concise, but (4/5)(2^{2016} -1) is also fine.But let's check if the problem expects a certain form. The problem states the general term as a_n=2^n cos(nπ/2). Since 2016 is a large exponent, perhaps expressing in terms of 2^{2016} might be preferable.But regardless, both forms are correct. Let me see if (2^{2018} -4)/5 simplifies further. Not really. Alternatively, factor 4/5*(2^{2016} -1). But again, both are correct.But since the problem is from a math competition or similar, usually they prefer the form without factoring out unnecessary constants. So probably (2^{2018} -4)/5 is the answer.Wait, but let me check with an example. For small n.Take n=2: S_2 = a1 +a2 =0 + (-4) =-4According to the formula: (2^{2*2 +2} -4)/5= (2^6 -4)/5=64-4=60/5=12. But S_2 is -4. Wait, that's not matching. Wait, something is wrong here.Wait, hold on. Wait, if n=2, then according to our formula, Sum_{k=1}^{1} (-4)^k = -4. So (4^{1+1} -4)/5= (16 -4)/5=12/5, which is not equal to -4. Wait, that contradicts. So where is the error here?Wait, wait, no. Wait, when n=2016, we had Sum_{k=1}^{1008} (-4)^k = (4^{1009} -4)/5. But if we take n=2, then the original sequence has 2 terms: a1 and a2. Then the sum S_2 =0 + (-4)= -4. But according to the formula when n=2, the number of even terms is 1. So Sum_{k=1}^1 (-4)^k = -4. Which equals (4^{1+1} -4)/5= (16 -4)/5=12/5=2.4, which is not equal to -4. So there's a discrepancy here.Wait, that suggests that my initial formula might be wrong. Wait, let me check. If Sum_{k=1}^m (-4)^k = (-4)*( (-4)^m -1 ) / (-4 -1 )For m=1: (-4)*( (-4)^1 -1 ) / (-5)= (-4)*(-4 -1)/-5= (-4)*(-5)/-5= 20/-5= -4. Correct.For m=1, the formula gives (-4*( (-4)^1 -1 )) / (-5)= (-4*(-5))/(-5)= 20/-5=-4. Correct.But when m=1, (4^{m+1} -4)/5=(4^2 -4)/5=(16-4)/5=12/5=2.4 which is not equal to -4. Therefore, the formula (4^{m+1} -4)/5 is not correct for m=1.But according to the previous steps, when m=1008, Sum_{k=1}^{1008} (-4)^k = ( -4*( (-4)^{1008} -1 )) / (-5 )= (-4*(4^{1008} -1 )) / (-5 )= (4*(4^{1008} -1 )) /5= (4^{1009} -4)/5Wait, but when m=1:Sum_{k=1}^1 (-4)^1= -4. But (4^{1+1} -4)/5=(16-4)/5=12/5, which is not -4. So discrepancy.Therefore, the formula (4^{m+1} -4)/5 is not correct for m=1. Therefore, my previous derivation must have an error.Wait, but where?Wait, let's rederive the sum again.Sum_{k=1}^m (-4)^k is a geometric series with first term a = -4, common ratio r = -4.Sum = a*(r^m -1)/(r -1 )So:Sum = (-4)*( (-4)^m -1 ) / ( (-4) -1 ) = (-4)*( (-4)^m -1 ) / (-5 )= [ (-4)*(-4)^m +4 ] / 5 = [4^{m+1} +4]/5 ?Wait, wait, let's do the algebra again.Sum = (-4)*( (-4)^m -1 ) / (-5 )= (-4)/(-5) * [ (-4)^m -1 ]= (4/5)*[ (-4)^m -1 ]But (-4)^m = (-1)^m *4^mSo Sum = (4/5)*[ (-1)^m*4^m -1 ]Which for even m:If m is even, (-1)^m=1, so Sum = (4/5)*(4^m -1 )If m is odd, (-1)^m=-1, so Sum=(4/5)*(-4^m -1 )But in our problem, m=1008, which is even. Therefore, Sum=(4/5)*(4^{1008} -1 )But earlier, we thought of m=1008, but Sum_{k=1}^{1008} (-4)^k = (4/5)*(4^{1008} -1 )But when m=1:Sum = (4/5)*( -4^1 -1 )=(4/5)*(-5)= -4, which is correct.Wait, so actually, the formula is (4/5)*( (-4)^m -1 )Which is equal to (4*(-4)^m -4)/5Therefore, for even m, since (-4)^m =4^m, so Sum= (4*4^m -4)/5=(4^{m+1} -4)/5But when m is even, 4^{m+1} -4 over 5.But in the case m=1 (odd), Sum=(4*(-4) -4)/5= (-16 -4)/5= -20/5= -4. Correct.But when m=2 (even):Sum= (-4)^1 + (-4)^2= -4 +16=12Formula: (4^{3} -4)/5=(64-4)/5=60/5=12. Correct.Therefore, when m is even, the formula (4^{m+1} -4)/5 works.But when m is odd, the formula gives (4*(-4)^m -4)/5, which for m odd is (4*(-4^m) -4)/5= (-4^{m+1} -4)/5, which for m=1 is (-16 -4)/5= -20/5=-4. Correct.Therefore, in our problem, since m=1008 is even, Sum_{k=1}^{1008} (-4)^k = (4^{1009} -4)/5Which is equivalent to (2^{2018} -4)/5, as 4^{1009}=2^{2018}Therefore, this is correct. Therefore, our initial calculation is correct. The m=1 test case was throwing me off because m=1 is odd, but in our problem m=1008 is even, so the formula (4^{m+1}-4)/5 applies.Therefore, the answer is (4^{1009} -4)/5 or (2^{2018} -4)/5. Both are correct. Depending on how the answer is expected.But let's check if the original problem has any particular form. The problem mentions a_n=2^n cos(nπ/2). Since the answer involves 2^{2018}, which is a very large number, but mathematically, either form is acceptable. However, the problem might prefer the answer in terms of 4^{1009}, but both are equivalent.Alternatively, since 4^{1009} is (2^2)^{1009}=2^{2018}, both forms are the same.So, to write the answer, either:boxed{dfrac{4^{1009} - 4}{5}} or boxed{dfrac{2^{2018} - 4}{5}}Both are correct. Let me check if the problem has a preference. Since the original general term is given in terms of 2^n, maybe expressing in terms of 2^{2018} is better. But either is acceptable. However, in math competitions, typically, they prefer simplified exponents, so 2^{2018} might be more straightforward. Let me confirm with an example.Suppose n=2, sum S_2= -4. According to formula (2^{2*2 +2} -4)/5=(2^6 -4)/5=60/5=12. Wait, but S_2=-4. Wait, something is wrong here. Wait, no:Wait, when n=2, S_2 is the sum of first 2 terms: a1 +a2=0 +(-4)= -4. But according to the formula (2^{2*2 +2}-4)/5=2^6=64, 64-4=60, 60/5=12. That's not correct.Wait, this inconsiderate! Wait, hold on. Wait, when n=2016, which is the total number of terms. But when we derived the formula, we considered m=1008 even terms. So the formula (2^{2018}-4)/5 is for the sum of 1008 terms (even terms from 1 to 2016). But if we take n=2, which corresponds to 1 even term (n=2), then the formula would be (2^{2*1 +2} -4)/5=(2^4 -4)/5= (16-4)/5=12/5=2.4, which is not equal to S_2=-4. So this seems conflicting.Wait, but wait, there's confusion between n in the original problem and m in the even terms. Let's clarify.Original problem: S_n is the sum of the first n terms. For n=2016, which is even, the number of non-zero terms is n/2=1008. So in general, for S_n where n is even, the sum is Sum_{k=1}^{n/2} (-4)^k. So for n=2, Sum_{k=1}^1 (-4)^k=-4. For n=4, Sum_{k=1}^2 (-4)^k= -4 +16=12.So for n=2, which is m=1 (number of even terms), the sum is (4^{m+1} -4)/5=(4^2 -4)/5=12/5=2.4 which is wrong. Wait, but according to previous correct calculation, Sum_{k=1}^m (-4)^k is (4/5)*( (-4)^m -1 )Therefore, for m=1: (4/5)*( (-4) -1 )=(4/5)*(-5)= -4. Correct.But when we expressed it as (4^{m+1} -4)/5, that is only valid when m is even. Wait, no.Wait, actually, the formula (4/5)*( (-4)^m -1 ) is general. If m is even, then (-4)^m=4^m, so Sum=(4/5)*(4^m -1 ). If m is odd, Sum=(4/5)*(-4^m -1 )Therefore, for general m, the sum is (4*(-4)^m -4)/5.But in our problem, m=1008, which is even, so Sum=(4^{1009}-4)/5.But when m=1008, (-4)^m=4^m, so yes.Therefore, the formula is correct.Therefore, for S_{2016}, the sum is (4^{1009}-4)/5, which is equal to (2^{2018}-4)/5.Therefore, the answer is boxed{dfrac{2^{2018} - 4}{5}}.Final Answerboxed{dfrac{2^{2018} - 4}{5}}</think>The general term of the sequence ({a_n}) is given by (a_n = 2^n cos frac{npi}{2}). We need to find the sum (S_{2016}) of the first 2016 terms of this sequence.1. Analyzing the sequence: - For odd (n), (cos frac{npi}{2} = 0), so (a_n = 0). - For even (n = 2k), (cos frac{2kpi}{2} = cos(kpi) = (-1)^k), so (a_{2k} = 2^{2k} cdot (-1)^k = (-4)^k).2. Sum of the sequence: - The first 2016 terms include 1008 even terms and 1008 odd terms (which are all zero). - Therefore, (S_{2016}) is the sum of the first 1008 even terms: (sum_{k=1}^{1008} (-4)^k).3. Geometric series sum: - The series (sum_{k=1}^{1008} (-4)^k) is a geometric series with first term (a = -4), common ratio (r = -4), and 1008 terms. - The sum of this geometric series is given by: [ S = frac{a(r^{n} - 1)}{r - 1} ] - Substituting the values, we get: [ S = frac{(-4) left((-4)^{1008} - 1right)}{-4 - 1} = frac{(-4) left(4^{1008} - 1right)}{-5} = frac{4 left(4^{1008} - 1right)}{5} ] - Simplifying further, we recognize (4^{1008} = (2^2)^{1008} = 2^{2016}) and (4^{1009} = 4 cdot 4^{1008} = 2^{2018}): [ S = frac{4^{1009} - 4}{5} = frac{2^{2018} - 4}{5} ]Thus, the value of (S_{2016}) is (boxed{dfrac{2^{2018} - 4}{5}}).]

🔑:Since a_n=2^{n}cos frac{nπ}{2}, where n in mathbb{N}^*, we can observe that:- For even terms, a_{2k} = 2^{n}cos kπ = 2^{n}(-1)^{k} = (-1)^{frac{n}{2}} cdot 2^{n}- For odd terms, a_{2k-1} = 2^{n}cos frac{(2k-1)π}{2} = 0, quad (k in mathbb{N}^*)Thus, the sum S_{2016} consists only of the even terms:S_{2016} = a_2 + a_4 + dots + a_{2n}S_{2016} = -2^2 + 2^4 - dots + 2^{2016}This is a geometric series with common ratio r = -4. We can apply the formula for the sum of a geometric series:S_n = frac{a_1 (1 - r^n)}{1 - r}S_{2016} = frac{-4 left[ (-4)^{1008} - 1 right]}{-4 - 1}S_{2016} = boxed{frac{4}{5}(2^{2016} - 1)}

🧠:[<think>Okay, so I need to find the general term formula for the sequence {a_n} where the recurrence relation is a_{n+2} = 3a_{n+1} - 2a_n, with initial terms a_1 = 1 and a_2 = 3. Hmm, let me think. I remember that linear recurrence relations can often be solved by finding the characteristic equation. Let me try that approach.First, the recurrence relation is a linear homogeneous recurrence relation with constant coefficients. The standard method for such problems is to assume a solution of the form a_n = r^n, where r is a constant. Substituting this into the recurrence relation should give a characteristic equation which we can solve for r.So, substituting a_n = r^n into the recurrence relation:a_{n+2} = 3a_{n+1} - 2a_n r^{n+2} = 3r^{n+1} - 2r^nDivide both sides by r^n (assuming r ≠ 0, which it should be since a_1 and a_2 are non-zero):r^2 = 3r - 2Now, rearranging terms gives the characteristic equation:r^2 - 3r + 2 = 0Let me solve this quadratic equation. The quadratic formula is r = [3 ± sqrt(9 - 8)] / 2 = [3 ± 1]/2, so the roots are:r = (3 + 1)/2 = 4/2 = 2 r = (3 - 1)/2 = 2/2 = 1So, the roots are r = 2 and r = 1. Since these are distinct real roots, the general solution to the recurrence relation is a linear combination of these roots raised to the nth power. That is,a_n = C(2^n) + D(1^n) = C(2^n) + DWhere C and D are constants to be determined by the initial conditions.Now, let's apply the initial conditions. The problem gives a_1 = 1 and a_2 = 3.First, plug in n = 1:a_1 = C(2^1) + D = 2C + D = 1Then, plug in n = 2:a_2 = C(2^2) + D = 4C + D = 3So now we have a system of equations:1) 2C + D = 1 2) 4C + D = 3To solve for C and D, subtract equation 1 from equation 2:(4C + D) - (2C + D) = 3 - 1 2C = 2 C = 1Now substitute C = 1 into equation 1:2(1) + D = 1 2 + D = 1 D = 1 - 2 D = -1Therefore, the general term formula is:a_n = 1*2^n + (-1)*1^n = 2^n - 1Let me check if this works with the given initial terms.For n = 1: 2^1 - 1 = 2 - 1 = 1, which matches a_1 = 1.For n = 2: 2^2 - 1 = 4 - 1 = 3, which matches a_2 = 3.Let's check n = 3 using the recurrence relation to be thorough. According to the formula, a_3 = 2^3 - 1 = 8 - 1 = 7.Using the recurrence relation: a_3 = 3a_2 - 2a_1 = 3*3 - 2*1 = 9 - 2 = 7. That matches.Similarly, a_4 = 3a_3 - 2a_2 = 3*7 - 2*3 = 21 - 6 = 15. Using the formula: 2^4 - 1 = 16 -1 = 15. Also matches.Seems like the formula works. Therefore, the general term is 2^n - 1.Final AnswerThe general term formula of the sequence is boxed{2^n - 1}.</think>To find the general term formula for the sequence ({a_n}) with the recurrence relation (a_{n+2} = 3a_{n+1} - 2a_n) and initial terms (a_1 = 1) and (a_2 = 3), we start by solving the characteristic equation.Assume a solution of the form (a_n = r^n). Substituting this into the recurrence relation gives:[r^{n+2} = 3r^{n+1} - 2r^n]Dividing both sides by (r^n) (assuming (r neq 0)):[r^2 = 3r - 2]Rearranging terms, we get the characteristic equation:[r^2 - 3r + 2 = 0]Solving this quadratic equation using the quadratic formula:[r = frac{3 pm sqrt{9 - 8}}{2} = frac{3 pm 1}{2}]The roots are:[r = 2 quad text{and} quad r = 1]The general solution to the recurrence relation is a linear combination of these roots:[a_n = C cdot 2^n + D cdot 1^n = C cdot 2^n + D]Using the initial conditions to determine the constants (C) and (D):For (n = 1):[2C + D = 1]For (n = 2):[4C + D = 3]Subtracting the first equation from the second:[2C = 2 implies C = 1]Substituting (C = 1) back into the first equation:[2(1) + D = 1 implies 2 + D = 1 implies D = -1]Thus, the general term formula is:[a_n = 2^n - 1]The final answer is (boxed{2^n - 1}).,<think>Okay, so I have this sequence problem here. The sequence is defined by a recurrence relation: a_{n+2} = 3a_{n+1} - 2a_n. They also give me the initial terms a₁ = 1 and a₂ = 3. I need to find the general term formula for this sequence. Hmm, recurrence relations. I remember something about characteristic equations from my discrete math class. Let me try to recall how that works.First, linear recurrence relations with constant coefficients can often be solved by finding the characteristic equation. The given recurrence is linear and homogeneous, right? The form is a_{n+2} - 3a_{n+1} + 2a_n = 0. So the characteristic equation should be r² - 3r + 2 = 0. Let me check that. Yeah, replacing a_{n+k} with r^k gives the characteristic equation. So, solving r² - 3r + 2 = 0.Factoring that quadratic equation: r² - 3r + 2 factors into (r - 1)(r - 2) = 0. So the roots are r = 1 and r = 2. Okay, so distinct real roots. Then the general solution should be a combination of the roots raised to the nth power. So the general term would be a_n = C(1)^n + D(2)^n. Simplifying that, since 1^n is just 1, so a_n = C + D(2^n).Now I need to find the constants C and D using the initial conditions. Let me plug in n = 1 and n = 2.For n = 1: a₁ = C + D(2^1) = C + 2D = 1.For n = 2: a₂ = C + D(2^2) = C + 4D = 3.So I have a system of equations:1) C + 2D = 12) C + 4D = 3Let me subtract equation 1 from equation 2 to eliminate C. (C + 4D) - (C + 2D) = 3 - 1 ⇒ 2D = 2 ⇒ D = 1.Then plug D = 1 back into equation 1: C + 2(1) = 1 ⇒ C = 1 - 2 ⇒ C = -1.So the general term should be a_n = -1 + 1*(2^n) = 2^n - 1.Let me check this formula with the initial terms to make sure.For n = 1: 2^1 - 1 = 2 - 1 = 1, which matches a₁ = 1.For n = 2: 2^2 - 1 = 4 - 1 = 3, which matches a₂ = 3.Let me check n = 3 to see if the recurrence holds. According to the formula, a₃ = 2^3 - 1 = 8 - 1 = 7. Using the recurrence relation: a₃ = 3a₂ - 2a₁ = 3*3 - 2*1 = 9 - 2 = 7. Perfect, that matches.Another term, n = 4: formula gives 2^4 -1 = 16 -1 =15. Recurrence: a₄ = 3a₃ -2a₂ = 3*7 -2*3 = 21 -6 =15. Also matches.So the formula seems to work. Therefore, the general term is a_n = 2^n -1.Wait, just to make sure I didn't make any mistakes in the algebra. The characteristic equation was correct. Roots at 1 and 2, leading to the general solution with constants C and D. Solved the system of equations correctly. Subtracting equations gives D=1, then C=-1. Plugging back, everything checks out. Terms computed for n=1,2,3,4 all satisfy both the formula and the recurrence. So I think that's correct.I don't see any errors in the steps. The key was recognizing the recurrence as a linear homogeneous recurrence with constant coefficients, forming the characteristic equation, finding the roots, writing the general solution, then using initial conditions to find the constants. All steps seem solid. Yep, I think that's the answer.Final AnswerThe general term formula of the sequence is boxed{2^n - 1}.</think>To find the general term formula for the sequence ({a_n}) defined by the recurrence relation (a_{n+2} = 3a_{n+1} - 2a_n) with initial conditions (a_1 = 1) and (a_2 = 3), we start by solving the characteristic equation associated with the recurrence relation.The characteristic equation for (a_{n+2} - 3a_{n+1} + 2a_n = 0) is:[r^2 - 3r + 2 = 0]Factoring this quadratic equation, we get:[(r - 1)(r - 2) = 0]The roots are (r = 1) and (r = 2). Therefore, the general solution to the recurrence relation is:[a_n = C cdot 1^n + D cdot 2^n = C + D cdot 2^n]Next, we use the initial conditions to determine the constants (C) and (D).For (n = 1):[a_1 = C + 2D = 1]For (n = 2):[a_2 = C + 4D = 3]We solve this system of equations:1. (C + 2D = 1)2. (C + 4D = 3)Subtracting the first equation from the second:[(C + 4D) - (C + 2D) = 3 - 1 implies 2D = 2 implies D = 1]Substituting (D = 1) back into the first equation:[C + 2 cdot 1 = 1 implies C = -1]Thus, the general term formula for the sequence is:[a_n = -1 + 2^n]Verifying with the initial terms and subsequent terms confirms the solution is correct. Therefore, the general term formula of the sequence is:[boxed{2^n - 1}]]

🔑:begin{array}{l}a_{n+2}-a_{n+1}=2left(a_{n+1}-a_{n}right) Rightarrow a_{n+1}-a_{n}=2^{n} Rightarrow a_{n}-a_{1}=2+2^{2}+cdots+2^{n-1} =2^{n}-2 Rightarrow a_{n}=2^{n}-1left(n in mathbf{N}_{+}right)end{array}The translation is as follows:begin{array}{l}a_{n+2}-a_{n+1}=2left(a_{n+1}-a_{n}right) Rightarrow a_{n+1}-a_{n}=2^{n} Rightarrow a_{n}-a_{1}=2+2^{2}+cdots+2^{n-1} =2^{n}-2 Rightarrow a_{n}=2^{n}-1left(n in mathbf{N}_{+}right)end{array}