🧠:[<think>Okay, let's see. I need to multiply 8 by 4 first. Hmm, 8 times 4. Let me think. 8 multiplied by 4 is like adding 8 four times: 8 + 8 + 8 + 8. Let me add them up. 8 plus 8 is 16, then another 8 makes 24, and one more 8 brings it to 32. So, 8 times 4 is 32. Got that part down.Now, the next step is to subtract 15 from that product. The product here is 32, right? So, subtracting 15 from 32. Let me do the math. 32 minus 15. Hmm, breaking it down: 32 minus 10 is 22, and then minus 5 more would be 17. Wait, is that right? Let me check another way. If I take 15 away from 32, starting from 15, how much do I need to add to get to 32? 15 plus 17 is 32. Yep, so 32 minus 15 is definitely 17. Let me make sure I didn't mix up any steps. First multiplication, then subtraction. Yeah, the problem says multiply first, then subtract. So 8*4=32, then 32-15=17. That seems correct. I don't think I made any mistakes here. The answer should be 17.</think>To solve the problem, first multiply 8 by 4: [8 times 4 = 32] Next, subtract 15 from the product: [32 - 15 = 17] Answer: boxed{17},<think>Okay, so I need to solve this math problem: Multiply 8 by 4, then subtract 15 from the product. Let me think through this step by step.First, the problem says to multiply 8 by 4. I remember that multiplication is basically repeated addition. So, 8 times 4 would be adding 8 four times: 8 + 8 + 8 + 8. Let me calculate that. 8 plus 8 is 16, then 16 plus 8 is 24, and then 24 plus 8 is 32. So, 8 multiplied by 4 equals 32. That seems right. Alternatively, I can just recall that 8 times 4 is one of those basic multiplication facts, which is 32. Okay, so the product here is 32.Now, the next part is to subtract 15 from that product. So, taking the result we just got, which is 32, and subtracting 15 from it. Let me do that subtraction. 32 minus 15. Hmm, let's break that down. If I subtract 10 from 32, that's straightforward: 32 - 10 = 22. Then I still need to subtract 5 more, so 22 - 5 = 17. Therefore, 32 minus 15 equals 17. Alternatively, I can think of 15 as 10 + 5 and subtract step by step. Either way, the result is 17.Wait, let me double-check that subtraction to be sure. Starting from 32, if I take away 15. Let's count backwards 15 from 32. So, subtract 10 to get to 22, then subtract 5 more. 22 minus 5 is indeed 17. Yep, that checks out. So the final answer should be 17.Putting it all together: 8 multiplied by 4 is 32, subtract 15 gives 17. No steps skipped here, seems straightforward. I don't think I made any mistakes in the calculation. Multiplication first, then subtraction, following the order of operations as mentioned in the problem. So the answer must be 17.</think>To solve the problem, first multiply 8 by 4:[8 times 4 = 32]Next, subtract 15 from the product:[32 - 15 = 17]The final answer is boxed{17}.]

🔑:8 cdot 4-15=17

🧠:[<think>Alright, let's tackle this problem. So, the question is about determining the minimum principal required to withdraw specific amounts indefinitely using compound interest. The withdrawals are 1 yuan at the end of the first year, 4 yuan at the end of the second year, 9 yuan at the end of the third year, and so on, up to n² yuan at the end of the nth year, and this pattern continues indefinitely. Hmm, okay.First, let me make sure I understand the problem correctly. We have a compound interest account with an annual interest rate i. The goal is to withdraw amounts that are perfect squares each year, starting with 1, then 4, then 9, etc., forever. The principal needs to be such that even after making these withdrawals indefinitely, the account never runs out of money. So, the minimum principal is the present value of all these future withdrawals. Right?Since it's compound interest, the present value (PV) of each withdrawal is the amount withdrawn divided by (1 + i) raised to the power of the number of years until that withdrawal. So, for the first withdrawal, which is 1 yuan at the end of year 1, the present value is 1/(1 + i). The second withdrawal is 4 yuan at the end of year 2, so its present value is 4/(1 + i)², and so on. Therefore, the total present value (which is the required principal) is the sum of n²/(1 + i)^n from n = 1 to infinity. That is, PV = Σ (n² / (1 + i)^n) for n = 1 to ∞.So, the problem reduces to calculating this infinite series. Now, I need to recall if there's a formula for the sum of n² x^n from n = 1 to infinity. I remember that for the sum of n x^n, the formula is x/(1 - x)², and for the sum of n² x^n, there is a similar formula. Let me try to remember or derive it.Let’s denote S = Σ n² x^n from n = 1 to ∞. To find S, I can start with the standard geometric series sum formula. The sum of x^n from n = 0 to ∞ is 1/(1 - x) for |x| < 1. Then, taking derivatives to bring down the powers of n. First derivative: d/dx Σ x^n = Σ n x^(n - 1) = 1/(1 - x)^2. Multiplying both sides by x gives Σ n x^n = x/(1 - x)^2. That's the sum for n x^n.Now, taking the derivative of Σ n x^n, which is d/dx [x/(1 - x)^2]. Let's compute that derivative:Let’s let f(x) = x/(1 - x)^2.f’(x) = [1*(1 - x)^2 - x*(-2)(1 - x)] / (1 - x)^4Simplify numerator:(1 - x)^2 + 2x(1 - x) = (1 - 2x + x²) + 2x - 2x² = 1 - 2x + x² + 2x - 2x² = 1 - x².Therefore, f’(x) = (1 - x²)/(1 - x)^4 = (1 - x)(1 + x)/(1 - x)^4) = (1 + x)/(1 - x)^3.But f’(x) is also the derivative of Σ n x^n, which is Σ n^2 x^{n - 1}. Therefore, multiplying by x gives Σ n^2 x^n = x*(1 + x)/(1 - x)^3.So, S = x(1 + x)/(1 - x)^3.So, the sum Σ n² x^n from n=1 to ∞ is x(1 + x)/(1 - x)^3.Therefore, in our case, x is 1/(1 + i). Because each term in the present value is n² / (1 + i)^n = n² x^n where x = 1/(1 + i).So, substituting x = 1/(1 + i) into the formula S = x(1 + x)/(1 - x)^3.Let me compute that step by step.First, compute x = 1/(1 + i).Then, compute 1 - x = 1 - 1/(1 + i) = i/(1 + i).Therefore, (1 - x)^3 = (i/(1 + i))^3.Then, compute 1 + x = 1 + 1/(1 + i) = (1 + i + 1)/(1 + i) = (2 + i)/(1 + i). Wait, is that right?Wait, 1 + x = 1 + [1/(1 + i)] = [(1 + i) + 1]/(1 + i) = (2 + i)/(1 + i). Yes, that's correct.Therefore, substituting into S:S = x(1 + x)/(1 - x)^3 = [1/(1 + i)] * [(2 + i)/(1 + i)] / [i/(1 + i)]^3Let me compute numerator and denominator separately.Numerator: [1/(1 + i)] * [(2 + i)/(1 + i)] = (2 + i)/[(1 + i)^2]Denominator: [i/(1 + i)]^3 = i^3 / (1 + i)^3Therefore, S = [ (2 + i)/(1 + i)^2 ] / [ i^3 / (1 + i)^3 ) ] = (2 + i)/(1 + i)^2 * (1 + i)^3 / i^3 ) = (2 + i)(1 + i)/i^3So, S = (2 + i)(1 + i)/i^3Let me expand (2 + i)(1 + i):2*1 + 2*i + i*1 + i*i = 2 + 2i + i + i² = 2 + 3i + i²But i is the annual interest rate, which is typically a small decimal, but maybe in mathematical terms, we can keep it as is.Therefore, S = (2 + 3i + i²)(1 + i)/i^3?Wait, no, actually, hold on. Wait, in the previous step:Wait, after multiplying numerator and denominator:Numerator: (2 + i)(1 + i)Denominator: i^3So, S = (2 + i)(1 + i)/i^3Yes, that's correct.Let me compute (2 + i)(1 + i):2*1 + 2*i + i*1 + i*i = 2 + 2i + i + i² = 2 + 3i + i²But since we can write S as (2 + 3i + i²)/i^3Alternatively, we can factor or rearrange terms.But perhaps leaving it as (2 + i)(1 + i)/i³ is simpler. Alternatively, expanding:(2 + i)(1 + i) = 2*1 + 2*i + i*1 + i*i = 2 + 2i + i + i² = 2 + 3i + i²But maybe that's not particularly helpful. Let's see.Alternatively, let's write everything in terms of i.Given that i is the interest rate, perhaps we can express S as (2 + i)(1 + i)/i³.Alternatively, maybe factor out terms:(2 + i)(1 + i) = 2(1 + i) + i(1 + i) = 2 + 2i + i + i² = same as before.Hmm.Alternatively, let me check if my steps are correct.So, starting from S = Σ n² x^n = x(1 + x)/(1 - x)^3.Then, substituting x = 1/(1 + i):S = [1/(1 + i)] * [1 + 1/(1 + i)] / [1 - 1/(1 + i)]³Simplify numerator and denominator:First, compute 1 + 1/(1 + i) = (1 + i + 1)/(1 + i) = (2 + i)/(1 + i)Then, compute 1 - 1/(1 + i) = (1 + i - 1)/(1 + i) = i/(1 + i)Therefore, denominator is [i/(1 + i)]³ = i³/(1 + i)³So, S = [1/(1 + i) * (2 + i)/(1 + i)] / [i³/(1 + i)³] = [ (2 + i)/(1 + i)² ] / [ i³/(1 + i)³ ] = (2 + i)/(1 + i)² * (1 + i)³/i³ = (2 + i)(1 + i)/i³Yes, that's correct.Therefore, S = (2 + i)(1 + i)/i³Alternatively, expanding the numerator:(2 + i)(1 + i) = 2(1 + i) + i(1 + i) = 2 + 2i + i + i² = 2 + 3i + i²So, S = (2 + 3i + i²)/i³But perhaps it's better to leave it as (2 + i)(1 + i)/i³ for simplicity.Alternatively, factor 2 + i and 1 + i. But maybe there's no need. So, the present value is S = (2 + i)(1 + i)/i³.But let me check with a simple example. Let's take i = 10%, so i = 0.1.Compute the present value using the formula and compare with summing the first few terms.Using the formula: S = (2 + 0.1)(1 + 0.1)/(0.1)^3 = (2.1)(1.1)/0.001 = 2.31 / 0.001 = 2310.Alternatively, compute the sum Σ n²/(1.1)^n from n=1 to infinity.But calculating that sum manually would be tedious, but let's check the first few terms:n=1: 1/(1.1) ≈ 0.9091n=2: 4/(1.1)^2 ≈ 4 / 1.21 ≈ 3.3058n=3: 9/(1.1)^3 ≈ 9 / 1.331 ≈ 6.7626n=4: 16/(1.1)^4 ≈ 16 / 1.4641 ≈ 10.928n=5: 25/(1.1)^5 ≈ 25 / 1.61051 ≈ 15.523Adding these up: 0.9091 + 3.3058 ≈ 4.2149; +6.7626 ≈ 10.9775; +10.928 ≈ 21.9055; +15.523 ≈ 37.4285. Continuing this, the terms keep increasing for a while before decreasing, but since it's n²/(1.1)^n, eventually the denominator grows faster. However, even after n=5, the sum is already 37.4285, and the formula gives 2310? That seems way too high. Wait, that can't be. There must be a mistake here.Wait, hold on. If i = 0.1, then 1/(1 + i) = 1/1.1 ≈ 0.9091. Then, according to the formula:S = (2 + 0.1)(1 + 0.1)/(0.1)^3 = 2.1 * 1.1 / 0.001 = 2.31 / 0.001 = 2310. That's 2310 yuan. But when we sum the first five terms, we get around 37.4285, and the total sum being 2310? That doesn't seem right because even though the series is infinite, the terms eventually become negligible. Wait, but n²/(1.1)^n decreases exponentially, so the sum should converge, but 2310 seems extremely large. Maybe the formula is correct? Let me check with another method.Alternatively, let's consider that for large n, the term n²/(1.1)^n tends to zero because the exponential decay outweighs the polynomial growth. However, the sum of all terms might indeed be large. Let's check with n=10:Term at n=10: 100/(1.1)^10 ≈ 100 / 2.5937 ≈ 38.554Sum after n=10: Previous sum was 37.4285 up to n=5. Then n=6: 36/(1.1)^6 ≈ 36 / 1.7716 ≈ 20.317; n=7: 49/(1.1)^7 ≈ 49 / 1.9487 ≈ 25.148; n=8: 64/(1.1)^8 ≈ 64 / 2.1436 ≈ 29.864; n=9: 81/(1.1)^9 ≈ 81 / 2.3579 ≈ 34.366; n=10: 100/(1.1)^10 ≈ 38.554.Adding these: 37.4285 (up to n=5) +20.317 ≈ 57.7455; +25.148 ≈82.8935; +29.864 ≈112.7575; +34.366 ≈147.1235; +38.554 ≈185.6775. So, after n=10, the sum is approximately 185.68. Still way below 2310. So, there's a contradiction here. That suggests that the formula is incorrect. Which means I must have made a mistake in the derivation.Wait, this is confusing. Let's double-check the formula for the sum of n² x^n.I thought it was x(1 + x)/(1 - x)^3. Let me confirm this.Looking it up, the standard formula for Σ_{n=1}^∞ n² x^n is x(1 + x)/(1 - x)^3. Yes, that is the correct formula for |x| < 1.So, substituting x = 1/(1 + i). Let's verify with i = 10%, x = 1/1.1 ≈0.9090909091.Then, S = x(1 + x)/(1 - x)^3.Compute 1 - x = 1 - 0.9090909091 ≈0.0909090909So, (1 - x)^3 ≈0.0909090909³ ≈0.0007513146Compute x(1 + x) ≈0.9090909091*(1 + 0.9090909091) ≈0.9090909091*1.9090909091 ≈1.73553719Then, S = 1.73553719 / 0.0007513146 ≈2310, which matches the earlier result. But when we sum the terms, even up to n=10, we only get around 185.68. The discrepancy is massive. So, either the formula is wrong, or my manual summation is missing something. Wait, but the formula is supposed to be correct. Maybe the problem is that the formula is for n²x^n, but in our case, x = 1/(1 + i) ≈0.9091, which is close to 1, so the convergence is very slow. Let me check the sum for larger n.But even at n=20, term n=20: 400/(1.1)^20 ≈400 / 6.7275 ≈59.44Sum up to n=20: previous sum at n=10 was ≈185.68. Then terms from n=11 to n=20:n=11: 121/(1.1)^11 ≈121 / 2.8531 ≈42.40n=12: 144/(1.1)^12 ≈144 / 3.1384 ≈45.88n=13: 169/(1.1)^13 ≈169 / 3.4523 ≈48.97n=14: 196/(1.1)^14 ≈196 / 3.7975 ≈51.61n=15: 225/(1.1)^15 ≈225 / 4.1772 ≈53.88n=16: 256/(1.1)^16 ≈256 / 4.5950 ≈55.69n=17: 289/(1.1)^17 ≈289 / 5.0545 ≈57.18n=18: 324/(1.1)^18 ≈324 / 5.5599 ≈58.27n=19: 361/(1.1)^19 ≈361 / 6.1159 ≈59.02n=20: 400/(1.1)^20 ≈400 / 6.7275 ≈59.44Adding these from n=11 to n=20:42.40 +45.88 =88.28; +48.97=137.25; +51.61=188.86; +53.88=242.74; +55.69=298.43; +57.18=355.61; +58.27=413.88; +59.02=472.90; +59.44=532.34.So total sum up to n=20 is previous 185.68 + 532.34 ≈718.02. But the formula says it should be 2310. So even at n=20, the sum is only 718.02, still way less than 2310. So, perhaps the convergence is extremely slow? Let me check for n=50.But this is getting tedious. Alternatively, maybe there's a mistake in the problem understanding. Wait, the problem says "to be able to withdraw in this manner indefinitely". So, the principal must be such that the withdrawals can continue forever, which mathematically means that the present value of all future withdrawals must equal the principal. Therefore, the formula should give the correct present value. The issue here is that when we take x = 1/(1 + i), which is 1/(1 + 0.1) ≈0.9091, and since |x| < 1, the series converges. However, the convergence is indeed slow because x is close to 1 when i is small. For i = 10%, x = 0.9091, which is quite close to 1, so the terms decay very slowly. Therefore, even at n=100, the term n²/(1.1)^n is still contributing a significant amount. Let's compute n=50.Term n=50: 2500/(1.1)^50. Compute (1.1)^50. Let's approximate. ln(1.1) ≈0.09531, so 50*ln(1.1) ≈4.7655, so exp(4.7655) ≈117.39. Therefore, (1.1)^50 ≈117.39, so term n=50 is 2500/117.39 ≈21.29. So, even at n=50, the term is still about 21.29, which is not negligible. Adding up all terms from n=1 to n=50 would accumulate a significant sum. However, even then, it's unclear if it would reach 2310. Alternatively, maybe my initial assumption is wrong. Wait, but the formula is derived mathematically, so it should hold. Let's check for a higher interest rate where x is smaller. Let's take i = 100%, so i = 1. Then x = 1/(1 + 1) = 0.5.Using the formula: S = (2 + 1)(1 + 1)/(1)^3 = (3)(2)/1 = 6.Calculating the sum manually:n=1: 1/2 = 0.5n=2: 4/4 = 1; sum=1.5n=3: 9/8 =1.125; sum=2.625n=4:16/16=1; sum=3.625n=5:25/32≈0.78125; sum≈4.40625n=6:36/64=0.5625; sum≈4.96875n=7:49/128≈0.3828; sum≈5.35155n=8:64/256=0.25; sum≈5.60155n=9:81/512≈0.1582; sum≈5.75975n=10:100/1024≈0.09766; sum≈5.8574Continuing:n=11:121/2048≈0.05908; sum≈5.9165n=12:144/4096≈0.03516; sum≈5.9517n=13:169/8192≈0.02063; sum≈5.9723n=14:196/16384≈0.01196; sum≈5.9843n=15:225/32768≈0.006866; sum≈5.9911And so on. Each term gets smaller, and the sum approaches 6. So, with i = 100%, the formula gives S = 6, and the sum converges to 6. So, in this case, the formula works. Therefore, when x is smaller (i is larger), the convergence is faster, and the formula holds. However, when i is small, like 10%, x is close to 1, and the convergence is very slow, requiring a large number of terms to approach the formula's result.Therefore, the formula is correct, and in the case of i = 10%, the sum is indeed 2310, even though the partial sums grow very slowly. This must be due to the nature of the series: even though each term eventually becomes small, the cumulative sum over an infinite horizon adds up to a large number.Therefore, going back to the original problem. The required principal is indeed S = (2 + i)(1 + i)/i³.But let me check if the formula is correctly applied. Yes, because substituting x = 1/(1 + i) into the sum formula S = x(1 + x)/(1 - x)^3 gives us the expression we derived.Therefore, the minimum principal required is (2 + i)(1 + i)/i³ yuan.But let me express this in a more simplified form. Expanding the numerator:(2 + i)(1 + i) = 2*1 + 2*i + i*1 + i*i = 2 + 2i + i + i² = 2 + 3i + i²So, S = (2 + 3i + i²)/i³Alternatively, factor out terms:= (i² + 3i + 2)/i³ = (i + 1)(i + 2)/i³Which is the same as before.Alternatively, we can split the fraction:= 2/i³ + 3/i² + 1/iBut that might not be helpful.Alternatively, write it as:= (1 + i)(2 + i)/i³So, depending on the preferred form. The answer can be written in different forms, but it's equivalent.Therefore, the minimal principal required is (1 + i)(2 + i)/i³.But let me check once again if this makes sense dimensionally or logically. The units should be yuan, and the formula involves i³ in the denominator, which for small i (like 0.05 or 5%), would make the denominator very small, hence the principal very large, which makes sense because a lower interest rate requires a larger principal to sustain perpetual withdrawals. Conversely, a higher i would result in a smaller principal. So, the formula behaves as expected.Therefore, after careful derivation and verification, the minimal principal required is indeed (1 + i)(2 + i)/i³ yuan.Final AnswerThe minimum principal required is boxed{dfrac{(1 + i)(2 + i)}{i^3}}.</think>To determine the minimum principal required to withdraw 1 yuan at the end of the first year, 4 yuan at the end of the second year, 9 yuan at the end of the third year, and so on indefinitely, we need to calculate the present value of all future withdrawals. The present value (PV) of each withdrawal is given by the formula:[text{PV} = sum_{n=1}^{infty} frac{n^2}{(1 + i)^n}]We use the formula for the sum of an infinite series (sum_{n=1}^{infty} n^2 x^n), which is known to be (frac{x(1 + x)}{(1 - x)^3}) for (|x| < 1). Here, (x = frac{1}{1 + i}).Substituting (x = frac{1}{1 + i}) into the formula, we get:[sum_{n=1}^{infty} n^2 left(frac{1}{1 + i}right)^n = frac{frac{1}{1 + i} left(1 + frac{1}{1 + i}right)}{left(1 - frac{1}{1 + i}right)^3}]Simplifying the numerator and denominator step by step:1. Compute (1 + frac{1}{1 + i}): [ 1 + frac{1}{1 + i} = frac{(1 + i) + 1}{1 + i} = frac{2 + i}{1 + i} ]2. Compute (1 - frac{1}{1 + i}): [ 1 - frac{1}{1 + i} = frac{(1 + i) - 1}{1 + i} = frac{i}{1 + i} ]3. Cube the denominator: [ left(frac{i}{1 + i}right)^3 = frac{i^3}{(1 + i)^3} ]Substituting these back into the formula:[sum_{n=1}^{infty} frac{n^2}{(1 + i)^n} = frac{frac{1}{1 + i} cdot frac{2 + i}{1 + i}}{frac{i^3}{(1 + i)^3}} = frac{frac{2 + i}{(1 + i)^2}}{frac{i^3}{(1 + i)^3}} = frac{(2 + i)(1 + i)}{i^3}]Thus, the minimum principal required is:[boxed{dfrac{(1 + i)(2 + i)}{i^3}}],<think>Okay, let's see. The problem is about finding the minimum principal required to allow withdrawing amounts that increase by the square of the year number each year, indefinitely, with compound interest. Hmm, so the withdrawals are 1 yuan at the end of the first year, 4 yuan at the end of the second year, 9 yuan at the end of the third year, and so on up to n² yuan at the end of the nth year, and this pattern repeats indefinitely. Wait, actually, maybe I misread that. Let me check again.Ah, the problem says "to withdraw 1 yuan at the end of the first year, 4 yuan at the end of the second year, ..., and n² yuan at the end of the nth year, and to be able to withdraw in this manner indefinitely." Wait, so it's not repeating after n years, but rather the withdrawals are 1, 4, 9, ..., n² each year, going on forever? That is, the withdrawal at the end of year k is k² yuan for each k = 1, 2, 3, ..., ad infinitum. So the problem is to find the present value of a perpetuity where each payment is k² at time k, for all k ≥ 1, with an annual interest rate i compounded annually.Yes, that makes sense. So the minimum principal required would be the present value of this infinite series of payments, each being the square of the year number. Therefore, the principal P must equal the sum from k=1 to infinity of k² / (1 + i)^k. So we need to compute the sum S = Σ_{k=1}^∞ [k² / (1 + i)^k]. Then, the minimum principal is S.But how do we compute this infinite sum? I remember that for the present value of a perpetuity with payments increasing arithmetically, there are formulas, but here the payments are increasing geometrically... Wait, no, they are increasing quadratically. Each payment is k squared. So perhaps we need a formula for the sum of k² x^k from k=1 to infinity, where x is 1/(1 + i). Then, once we have that, we can plug in x = 1/(1 + i) to get the present value.Right. So let me recall the formula for the sum of k² x^k. I think there's a standard result for this. Let me try to derive it.First, let's denote S = Σ_{k=1}^∞ k² x^k.We know that for |x| < 1, the sum Σ_{k=0}^∞ x^k = 1/(1 - x). Then, taking derivatives, we can find sums involving k and k².First, let's recall that:Σ_{k=1}^∞ k x^{k} = x / (1 - x)^2.That's the first derivative. Then, for the sum involving k², we can take the derivative again.Let me compute it step by step.Let’s start with S1 = Σ_{k=1}^∞ k x^{k} = x / (1 - x)^2.Then, take the derivative of S1 with respect to x:dS1/dx = Σ_{k=1}^∞ k^2 x^{k - 1} = (1 + x) / (1 - x)^3.Wait, let me check that. Wait, S1 = x / (1 - x)^2. So derivative of S1 is [1*(1 - x)^2 - x*2*(1 - x)*(-1)] / (1 - x)^4. Let me compute that:d/dx [x / (1 - x)^2] = [1*(1 - x)^2 + x*2*(1 - x)] / (1 - x)^4Wait, applying the quotient rule: derivative of numerator (1) times denominator minus numerator times derivative of denominator, over denominator squared. Wait, no. Wait, the function is x divided by (1 - x)^2. So the derivative is [1*(1 - x)^2 - x*(-2)(1 - x)] / (1 - x)^4.Wait, more carefully:If f(x) = x / (1 - x)^2, then f’(x) = [1*(1 - x)^2 + x*2*(1 - x)] / (1 - x)^4. Wait, no, hold on. The quotient rule is [f’(x)g(x) - f(x)g’(x)] / [g(x)]^2.But here, f(x) = x, g(x) = (1 - x)^2. So f’(x) = 1, g’(x) = 2(1 - x)*(-1) = -2(1 - x). Therefore, f’(x)g(x) - f(x)g’(x) = 1*(1 - x)^2 - x*(-2)(1 - x) = (1 - x)^2 + 2x(1 - x).Factor out (1 - x):(1 - x)[(1 - x) + 2x] = (1 - x)(1 + x).Thus, f’(x) = [(1 - x)(1 + x)] / (1 - x)^4) = (1 + x) / (1 - x)^3.Therefore, d/dx [x / (1 - x)^2] = (1 + x) / (1 - x)^3.But also, d/dx S1 = d/dx Σ_{k=1}^∞ k x^{k} = Σ_{k=1}^∞ k^2 x^{k - 1} = (1 + x) / (1 - x)^3.Therefore, multiplying both sides by x:Σ_{k=1}^∞ k^2 x^{k} = x(1 + x) / (1 - x)^3.So, the sum S = Σ_{k=1}^∞ k² x^k = x(1 + x) / (1 - x)^3.Hence, in our problem, x is 1/(1 + i), so substituting x = 1/(1 + i), we get:S = [1/(1 + i) * (1 + 1/(1 + i))] / [1 - 1/(1 + i)]^3.Let me compute each part step by step.First, compute x = 1/(1 + i). Then:Numerator: x(1 + x) = [1/(1 + i)] * [1 + 1/(1 + i)] = [1/(1 + i)] * [(1 + i + 1)/(1 + i)] = [1/(1 + i)] * [(2 + i)/(1 + i)] = (2 + i)/[(1 + i)^2].Denominator: [1 - x]^3 = [1 - 1/(1 + i)]^3 = [(1 + i - 1)/(1 + i)]^3 = [i/(1 + i)]^3 = i³ / (1 + i)^3.Therefore, S = [ (2 + i)/ (1 + i)^2 ] / [ i³ / (1 + i)^3 ] ) = [ (2 + i)/ (1 + i)^2 ] * [ (1 + i)^3 / i³ ] ) = (2 + i)(1 + i) / i³.Simplify numerator: (2 + i)(1 + i) = 2(1 + i) + i(1 + i) = 2 + 2i + i + i² = 2 + 3i + i². But since i is the interest rate, we can leave it as (2 + i)(1 + i) for now.Alternatively, compute it:(2 + i)(1 + i) = 2*1 + 2*i + i*1 + i*i = 2 + 2i + i + i² = 2 + 3i + i².But in finance, usually, i is a small decimal, but in mathematical terms, we can keep it as is. However, perhaps the answer is expected in terms of (2 + i)(1 + i)/i³. Wait, but let's check the calculation again.Wait, let's go through the steps once more to verify.Original formula: S = x(1 + x)/(1 - x)^3.Substitute x = 1/(1 + i):Numerator:x(1 + x) = (1/(1 + i))(1 + 1/(1 + i)) = (1/(1 + i)) * ((1 + i) + 1)/(1 + i)) = (1/(1 + i)) * (2 + i)/(1 + i) = (2 + i)/ (1 + i)^2.Denominator:(1 - x)^3 = (1 - 1/(1 + i))^3 = ((1 + i - 1)/(1 + i))^3 = (i/(1 + i))^3.Therefore, S = [ (2 + i)/(1 + i)^2 ] / [ (i/(1 + i))^3 ] = [ (2 + i)/(1 + i)^2 ] * [ (1 + i)^3 / i^3 ] = (2 + i)(1 + i) / i^3.Yes, so that's S = (2 + i)(1 + i)/i³.But perhaps we can expand (2 + i)(1 + i):(2 + i)(1 + i) = 2*1 + 2*i + i*1 + i*i = 2 + 2i + i + i² = 2 + 3i + i².Therefore, S = (2 + 3i + i²)/i³.But in financial contexts, it's often expressed in terms of the interest rate. However, unless there's a simplification here, that might be the final answer.Alternatively, maybe we can factor numerator and denominator. Let's see.Wait, if we factor numerator:2 + 3i + i² = i² + 3i + 2 = (i + 1)(i + 2). Hmm, interesting, that's the same as before. So (i + 1)(i + 2) = i² + 3i + 2. Therefore, S = (i + 1)(i + 2)/i³.Alternatively, written as (1 + i)(2 + i)/i³.But depending on the form required, maybe we can write it differently.Alternatively, factor out (1 + i):(1 + i)(2 + i)/i³.But perhaps the problem expects a different form. Let me check with an example. Suppose i is 10%, so 0.1. Then, compute S:Numerator: (1 + 0.1)(2 + 0.1) = 1.1 * 2.1 = 2.31.Denominator: 0.1³ = 0.001.Thus, S = 2.31 / 0.001 = 2310. So if the interest rate is 10%, the principal required is 2310 yuan. Let's verify with the sum.Wait, but that seems high. Let me compute the sum numerically for i = 0.1.The present value should be Σ_{k=1}^∞ k² / (1.1)^k.Compute the first few terms:At k=1: 1 / 1.1 ≈ 0.9091k=2: 4 / 1.21 ≈ 3.3058k=3: 9 / 1.331 ≈ 6.7627k=4: 16 / 1.4641 ≈ 10.9282k=5: 25 / 1.61051 ≈ 15.523k=6: 36 / 1.7721 ≈ 20.315k=7: 49 / 1.9487 ≈ 25.145k=8: 64 / 2.1436 ≈ 29.856k=9: 81 / 2.3579 ≈ 34.363k=10: 100 / 2.5937 ≈ 38.554Adding these up:0.9091 + 3.3058 = 4.2149+6.7627 = 10.9776+10.9282 = 21.9058+15.523 ≈ 37.4288+20.315 ≈ 57.7438+25.145 ≈ 82.8888+29.856 ≈ 112.7448+34.363 ≈ 147.1078+38.554 ≈ 185.6618So after 10 years, the present value is already 185.66. But our formula gave 2310, which is way higher. Wait, that can't be. There must be a mistake here.Wait, hold on, when i=0.1, x=1/(1 + i)=1/1.1≈0.9091. So our formula S = x(1 + x)/(1 - x)^3.Let me compute that:x = 1/1.1 ≈ 0.9090909091x(1 + x) = 0.9090909091*(1 + 0.9090909091) ≈ 0.9090909091*1.9090909091 ≈ 1.73553719(1 - x) = 1 - 0.9090909091 ≈ 0.0909090909(1 - x)^3 ≈ (0.0909090909)^3 ≈ 0.0007513148So S ≈ 1.73553719 / 0.0007513148 ≈ 2309.4, which aligns with the previous result. However, when we computed the sum up to k=10, we only got about 185.66, which is much less than 2309.4. This suggests that the formula is correct because the sum converges to 2309.4 as k approaches infinity. The discrepancy is because the terms keep adding up even for large k. Let's check a term when k=100.At k=100, the term is 10000 / (1.1)^100. Let's compute (1.1)^100. Taking log, ln(1.1^100)=100*ln(1.1)≈100*0.09531≈9.531, so e^9.531≈13, But wait, e^9.531 is approximately 13, but actually, (1.1)^100 is approximately 13780. So 10000 / 13780 ≈ 0.7256. So even at k=100, the term is about 0.7256. So the sum from k=1 to infinity is indeed adding up to around 2309, with contributions from all terms, even though they get smaller. But each term k² / (1.1)^k eventually tends to zero, but does so slowly. So the formula is correct.Therefore, the formula S = (1 + i)(2 + i)/i³ is correct. Wait, but in our substitution above, S was equal to (2 + i)(1 + i)/i³. Let me check again:Original variables:S = x(1 + x)/(1 - x)^3, where x = 1/(1 + i).Therefore:x = 1/(1 + i)1 + x = 1 + 1/(1 + i) = (1 + i + 1)/(1 + i) = (2 + i)/(1 + i)Wait, so x(1 + x) = [1/(1 + i)] * [(2 + i)/(1 + i)] = (2 + i)/(1 + i)^2Then, denominator (1 - x)^3 = [i/(1 + i)]^3 = i³/(1 + i)^3Thus, S = (2 + i)/(1 + i)^2 divided by i³/(1 + i)^3) = (2 + i)/(1 + i)^2 * (1 + i)^3/i³) = (2 + i)(1 + i)/i³.Yes, that's correct.Alternatively, we can write this as:(1 + i)(2 + i)/i³.Alternatively, expanding numerator:(1 + i)(2 + i) = 2 + 3i + i².Therefore, S = (2 + 3i + i²)/i³.But in terms of financial notation, sometimes it's preferred to have it in terms of i. However, perhaps the problem expects the answer in terms of (1 + i)(i + 2)/i³. Alternatively, factor 1 + i:But 1 + i is just (1 + i), so perhaps leave it as (1 + i)(2 + i)/i³.Alternatively, divide numerator and denominator by i³, but I don't think that simplifies.Alternatively, write as:[ (1 + i)(2 + i) ] / i³.Alternatively, expand numerator:2(1 + i) + i(1 + i) = 2 + 2i + i + i² = 2 + 3i + i².But unless there's a standard formula, this is as simplified as it gets.Wait, but maybe we can relate this to annuity formulas. Let me recall that the present value of an increasing perpetuity where payments are 1, 2, 3, ..., is 1/i² + 1/i. Wait, is that right?Wait, actually, the present value of a perpetuity with payments 1, 2, 3, ... at the end of each year is (1 + i)/i². Wait, let's confirm:The present value is Σ_{k=1}^∞ k/(1 + i)^k. Using the formula for Σ k x^k with x = 1/(1 + i):Σ k x^k = x/(1 - x)^2. Therefore, substituting x = 1/(1 + i):PV = [1/(1 + i)] / [1 - 1/(1 + i)]² = [1/(1 + i)] / [i/(1 + i)]² = [1/(1 + i)] * [(1 + i)^2 / i²] = (1 + i)/i².So yes, that's the standard formula: (1 + i)/i² for the present value of a perpetuity increasing by 1 each year.Similarly, for our case, the payments are k², so the present value is given by S = (1 + i)(2 + i)/i³.Alternatively, if we consider that the present value for k is (1 + i)/i², then for k², perhaps there's a higher-order derivative involved. But as we derived earlier, using calculus, we get S = (1 + i)(2 + i)/i³.Therefore, the minimum principal required is (1 + i)(2 + i)/i³ yuan.Alternatively, written as (2 + i)(1 + i)/i³.But perhaps the answer is presented differently. Let me check with another approach.Alternative Approach:Let me consider that the present value of the perpetuity is S = Σ_{k=1}^∞ k² v^k, where v = 1/(1 + i).We can use the formula for the sum of k² v^k.As derived before, Σ_{k=1}^∞ k² v^k = v(1 + v)/(1 - v)^3.But since v = 1/(1 + i), we can express this in terms of i:1 - v = 1 - 1/(1 + i) = i/(1 + i).Therefore, 1 - v = i/(1 + i).Hence, 1 - v = d, where d is the discount rate, but perhaps not necessary here.Therefore, substituting:Σ_{k=1}^∞ k² v^k = v(1 + v)/( (i/(1 + i))^3 ) = [1/(1 + i) * (1 + 1/(1 + i)) ] / [i³/(1 + i)^3] = [ (2 + i)/(1 + i)^2 ] / [i³/(1 + i)^3 ] = (2 + i)/(1 + i)^2 * (1 + i)^3 / i³ = (2 + i)(1 + i)/i³, which is the same result.Therefore, regardless of the approach, we arrive at S = (1 + i)(2 + i)/i³.Hence, the minimum principal required is (1 + i)(2 + i)/i³ yuan.But let me check with an example from calculus. Suppose we have S = Σ_{k=1}^∞ k² x^k = x(1 + x)/(1 - x)^3. Then, substituting x = 1/(1 + i) gives S = [1/(1 + i) * (1 + 1/(1 + i))]/[1 - 1/(1 + i)]^3. Which simplifies to the same expression.Therefore, the answer is confirmed.Alternatively, perhaps to write it in terms of the interest rate as:S = (1 + i)(2 + i)/i³.Alternatively, factor 1 + i and write:S = (1 + i)(i + 2)/i³.But regardless, this seems to be the simplified form.Therefore, the minimum principal required is boxed{dfrac{(1 + i)(2 + i)}{i^3}}.Final AnswerThe minimum principal required is boxed{dfrac{(1 + i)(2 + i)}{i^3}}.</think>To find the minimum principal required to withdraw 1 yuan at the end of the first year, 4 yuan at the end of the second year, ..., and ( n^2 ) yuan at the end of the ( n )-th year indefinitely, we need to compute the present value of an infinite series of payments where each payment at the end of year ( k ) is ( k^2 ) yuan. The present value ( S ) of this series is given by the sum:[S = sum_{k=1}^{infty} frac{k^2}{(1 + i)^k}]Using the formula for the sum of ( k^2 x^k ) where ( x = frac{1}{1 + i} ), we have:[sum_{k=1}^{infty} k^2 x^k = frac{x(1 + x)}{(1 - x)^3}]Substituting ( x = frac{1}{1 + i} ):1. Compute the numerator: [ x(1 + x) = frac{1}{1 + i} left(1 + frac{1}{1 + i}right) = frac{1}{1 + i} cdot frac{2 + i}{1 + i} = frac{2 + i}{(1 + i)^2} ]2. Compute the denominator: [ (1 - x)^3 = left(1 - frac{1}{1 + i}right)^3 = left(frac{i}{1 + i}right)^3 = frac{i^3}{(1 + i)^3} ]3. Combine the results: [ S = frac{frac{2 + i}{(1 + i)^2}}{frac{i^3}{(1 + i)^3}} = frac{(2 + i)(1 + i)}{i^3} ]Thus, the minimum principal required is:[boxed{dfrac{(1 + i)(2 + i)}{i^3}}]]

🔑:To solve for the principal amount that will amount to 1 yuan (principal and interest) after n years, the principal should be (1+i)^{-n} yuan; to amount to n^{2} yuan (principal and interest), the principal should be n^{2}(1+i)^{-n} yuan. Therefore, the total principal is sum_{n=1}^{infty} n^{2}(1+i)^{-n}.Clearly, the above expression can be viewed as the generating function of the sequence left{n^{2}right} when x=frac{1}{1+i}. The generating function of the sequence left{n^{2}right} isbegin{aligned}f(x) & =0 cdot 1+1^{2} cdot x+2^{2} cdot x^{2}+3^{2} cdot x^{3}+cdots+n^{2} x^{n}+cdots & =x+sum_{n=2}^{infty} n^{2} x^{n}=x+sum_{n=2}^{infty}left(C_{n+1}^{2}+C_{n}^{2}right) x^{n} & =x+sum_{n=2}^{infty} C_{n+1}^{2} x^{n}+sum_{n=2}^{infty} C_{n}^{2} x^{n}=sum_{n=1}^{infty} C_{n+1}^{2} x+sum_{n=2}^{infty} C_{n}^{2} x^{n} & =sum_{n=0}^{infty} C_{n+2}^{2} x^{n+1}+sum_{n=0}^{infty} C_{n+2}^{2} x^{n+2} & =frac{x}{(1-x)^{3}}+frac{x^{2}}{(1-x)^{3}}=frac{x(1+x)}{(1-x)^{3}} .end{aligned}Noting that x=frac{1}{1+i}, hence sum_{n=1}^{infty} n^{2}(1+i)^{-n}=frac{(1+i)(2+i)}{i^{3}}.

🧠:[<think>Okay, let's see. I need to find the sine of the angle α that a certain plane makes with all 12 edges of a cube. The problem states that the plane is inclined at angle α to each of the cube's edges. Hmm, interesting. So first, let me visualize a cube. A cube has 12 edges: 12 edges, 8 vertices, 6 faces. Each edge is connected to two vertices, and each face is a square.Now, the plane in question makes the same angle α with all 12 edges. That seems a bit tricky because the edges of a cube are oriented along three different axes: x, y, z. Each edge is aligned with one of these three axes. So, if I think of a standard cube in 3D space with edges parallel to the coordinate axes, then there are edges along the x-axis, y-axis, and z-axis. Each edge direction is orthogonal to the others. But the plane makes the same angle with all of them. That suggests the plane is somehow symmetrically placed with respect to the cube's edges. Maybe it's equally inclined to all three axes? If that's the case, then the normal vector of the plane would have components equal in some way. For example, if the normal vector is (1,1,1), which is equally inclined to all three axes, then the plane equation would be x + y + z = d for some constant d. But wait, does a plane with normal vector (1,1,1) make equal angles with all the edges?Wait, the angle between a line and a plane is defined as the complement of the angle between the line and the normal to the plane. Specifically, the angle α between the line and the plane is equal to 90 degrees minus the angle between the line and the normal vector. So, if θ is the angle between the line (edge) and the normal vector, then sin α = sin(90° - θ) = cos θ. Therefore, sin α = cos θ, where θ is the angle between the edge and the normal vector.Therefore, since all edges are along the coordinate axes, each edge is aligned with one of the standard basis vectors (i, j, k). Let's denote the normal vector to the plane as n = (a, b, c). Then, the angle θ between the normal vector and, say, the x-axis (edge along x-axis) is given by the dot product:cos θ_x = (n ⋅ i) / (|n| |i|) = a / sqrt(a² + b² + c²)Similarly, cos θ_y = b / sqrt(a² + b² + c²), and cos θ_z = c / sqrt(a² + b² + c²)But the angle α between the plane and the edge is such that sin α = cos θ. Since the problem states that sin α is the same for all edges, which are along x, y, z axes. Therefore, cos θ_x = cos θ_y = cos θ_z. Therefore, the normal vector must make equal angles with all three coordinate axes. Therefore, the direction cosines of the normal vector are equal. Therefore, a = b = c. So, the normal vector is of the form (a, a, a). So, we can take n = (1, 1, 1) for simplicity.Therefore, the normal vector is (1,1,1). Then, the angle θ between the normal vector and any of the coordinate axes is θ = arccos(1 / sqrt(3)). Therefore, sin α = cos θ = 1 / sqrt(3). Therefore, sin α = 1/√3, which simplifies to √3/3.Wait, hold on. Let me check that again. The angle α is between the edge (which is along a coordinate axis) and the plane. The formula is sin α = cos θ, where θ is the angle between the edge and the normal. So, if θ is arccos(1/√3), then sin α = 1/√3, which is approximately 0.577. So, sin α = 1/√3, which is equal to √3/3. Yes, because 1/√3 is rationalized as √3/3.But let me verify this. Let's take a specific example. Suppose the cube is aligned with the coordinate axes, edges along x, y, z. Let's take the plane x + y + z = 0. The normal vector is (1,1,1). Then, the angle between the x-axis and the normal vector is θ, where cos θ = 1 / sqrt(3). So, θ = arccos(1/√3). Therefore, the angle between the x-axis and the plane is α = 90° - θ. Then, sin α = sin(90° - θ) = cos θ = 1/√3. So yes, sin α = 1/√3. Therefore, sin α = √3/3.But wait, is the plane x + y + z = 0 the only such plane? Or are there other planes that make the same angle with all edges? For example, if we take a different normal vector, but with a = b = c, then scaling the normal vector doesn't change the direction, so any scalar multiple would give the same angles. So, the normal vector has to be in the direction (1,1,1) or (-1,1,1), etc., but as long as the magnitudes are equal, the direction cosines will be equal. So, yes, the plane must have a normal vector with equal components. Therefore, this should be the case.But let me check if this is correct. Let's compute the angle between the plane and an edge. Let's take the edge along the x-axis. The direction vector of the edge is i = (1,0,0). The normal vector of the plane is (1,1,1). The angle between the edge and the normal is θ, so:cos θ = (i ⋅ n) / (|i| |n|) = (1*1 + 0*1 + 0*1) / (1 * sqrt(1+1+1)) = 1 / sqrt(3)So θ = arccos(1/√3). Then, the angle between the edge and the plane is α = 90° - θ. Therefore, sin α = sin(90° - θ) = cos θ = 1/√3. Therefore, sin α = 1/√3.Similarly, for the edges along y and z axes, the calculation is the same. Since the normal vector has equal components, the angle between the normal and each axis is the same, so sin α is the same for all edges. Therefore, the answer is 1/√3, which is √3/3. So, sin α = √3/3. Therefore, the answer should be √3/3.But let me verify with another approach. Suppose we have a cube with edge length 1. Let's consider one edge, say along the x-axis from (0,0,0) to (1,0,0). The plane in question makes an angle α with this edge. Similarly, it makes angle α with all other edges.The angle between a line and a plane is the angle between the line and its orthogonal projection onto the plane. Alternatively, it's the complement of the angle between the line and the normal to the plane, as we discussed before.Alternatively, we can parametrize the problem. Let's suppose the plane is given by ax + by + cz = d. The normal vector is (a, b, c). The angle α between the plane and any edge (which is along a coordinate axis) is given by sin α = |projection of the edge direction onto the plane| divided by |edge direction|. Wait, no. The angle between the line and the plane is defined as the angle between the line and its projection onto the plane. So, sin α is equal to the sine of that angle. Alternatively, it can be computed as the component of the direction vector perpendicular to the plane divided by the magnitude of the direction vector.Wait, perhaps another way: the angle α between the line and the plane satisfies sin α = |n ⋅ v| / |n||v|, where n is the normal vector and v is the direction vector of the line. Wait, no, that's the angle between the line and the normal. So, sin α = sin(90° - θ) = cos θ, where θ is the angle between the line and the normal. So yes, as we had before, sin α = cos θ = |n ⋅ v| / |n||v|. Since the direction vector v is a unit vector along the axis, |v| = 1. Therefore, sin α = |n ⋅ v| / |n|.But in our case, all edges are along the coordinate axes, so their direction vectors are i, j, k. Therefore, for each edge direction v (i, j, or k), sin α = |n ⋅ v| / |n|. Since we need sin α to be the same for all edges, then |a| / |n| = |b| / |n| = |c| / |n|. Therefore, |a| = |b| = |c|. So, the normal vector must have components with equal absolute values. Therefore, the normal vector is of the form (±1, ±1, ±1). Therefore, regardless of the signs, the magnitude of each component is the same. Therefore, |a| = |b| = |c|. Therefore, sin α = |a| / sqrt(a² + b² + c²) = |a| / sqrt(3a²) ) = |a| / (|a| sqrt(3)) ) = 1 / sqrt(3). Therefore, sin α = 1/√3. Which is √3/3. So that's consistent with the previous result.Therefore, regardless of the signs of the normal vector components, as long as their absolute values are equal, sin α will be 1/√3. Therefore, the answer is √3 over 3, which is 1 over √3.Therefore, the answer is √3/3.But let me check once more with coordinates. Let's take the plane x + y + z = 0. Take an edge along the x-axis from (0,0,0) to (1,0,0). The direction vector is (1,0,0). The normal vector is (1,1,1). The angle between the direction vector and the normal vector is θ, with cos θ = (1*1 + 0*1 + 0*1)/sqrt(1^2 + 1^2 + 1^2)*sqrt(1^2 + 0^2 + 0^2)) = 1 / sqrt(3). Therefore, θ = arccos(1/√3). Then, the angle between the direction vector and the plane is α = 90° - θ. Therefore, sin α = sin(90° - θ) = cos θ = 1/√3. Therefore, sin α = 1/√3 = √3/3. Similarly, for the edge along the y-axis, direction vector (0,1,0), the dot product with normal is 1, same as before, so same angle. Same for z-axis. So all edges along the coordinate axes have the same angle α with the plane. Then, what about the edges that are not along the coordinate axes? Wait, wait, hold on. Wait, the cube edges are only along the coordinate axes. Wait, a cube has edges along x, y, z axes. Each edge is aligned with one axis. So, in total, 12 edges: 4 along x, 4 along y, 4 along z. But in a unit cube, each axis has 4 edges? Wait, no. Wait, a cube has 12 edges: for each of the three axes, there are 4 edges. Wait, no. Let's think. A cube has 12 edges: 12 edges. Each face has 4 edges, but each edge is shared by two faces. So, total edges: 6 faces * 4 edges / 2 = 12 edges. Each edge is parallel to one of the three coordinate axes. Therefore, for each axis, there are 4 edges parallel to that axis. So, in total, 4 edges along x, 4 along y, 4 along z. Therefore, all edges are aligned with the coordinate axes. Therefore, the 12 edges are all along x, y, or z. Therefore, if the plane makes angle α with all edges, then as we computed, sin α = 1/√3. Therefore, the answer is √3/3.Wait, but hold on. The problem says "a plane that makes an angle α with all 12 edges of a cube". So, all 12 edges. But in our calculation, we considered edges along the coordinate axes, but in reality, each edge is along one of the coordinate axes. Therefore, all edges are either along x, y, or z. Therefore, as we saw, the angle between the plane and each edge is α, so sin α = 1/√3. Therefore, the answer is √3/3. Therefore, sin α = √3/3.Alternatively, let me think of another way. Let's consider the cube with vertices at (0,0,0) to (1,1,1). Let’s take the plane x + y + z = 0. Wait, but that plane doesn't pass through the cube. The cube is from (0,0,0) to (1,1,1), so the plane x + y + z = 0 cuts through the origin but doesn't intersect the cube except at the origin. Maybe a different plane, like x + y + z = 1. Let's see. That plane would intersect the cube at points (1,0,0), (0,1,0), and (0,0,1). So, forming a triangle on the cube. But does this plane make the same angle with all edges?Wait, perhaps it's easier to not focus on a specific plane but rather use the general result we found earlier. Since the normal vector must have equal components, so the plane equation is x + y + z = d, and the angle calculation is the same regardless of d, because shifting the plane doesn't change the angles it makes with the edges. The angle only depends on the orientation (normal vector), not the position. Therefore, even if the plane is shifted, the angles with the edges remain the same. Therefore, our previous calculation holds, and sin α = 1/√3.Therefore, I think the answer is √3/3. But let me check once more with another method. Let's use vector projections.Suppose we have a cube with edges along the coordinate axes. Let’s take an edge along the x-axis, say from (0,0,0) to (1,0,0). The direction vector of this edge is v = (1,0,0). Let the plane have normal vector n = (a,b,c). The angle α between the edge and the plane is given by α = 90° - θ, where θ is the angle between v and n. Therefore, sin α = cos θ.cos θ = (v ⋅ n) / (|v| |n|) = a / sqrt(a² + b² + c²).Therefore, sin α = a / sqrt(a² + b² + c²). Similarly, for edges along y and z axes, sin α = b / sqrt(a² + b² + c²) and sin α = c / sqrt(a² + b² + c²) respectively.Since all these sines must be equal, we have a / sqrt(a² + b² + c²) = b / sqrt(a² + b² + c²) = c / sqrt(a² + b² + c²). Therefore, a = b = c. Therefore, the normal vector is (a,a,a), so we can write it as (1,1,1) after scaling. Therefore, sin α = 1 / sqrt(1² + 1² + 1²) = 1 / sqrt(3) = √3 / 3.Therefore, this confirms the result. Therefore, the answer is √3 / 3.But let me check if there is another possible configuration. For example, if the plane is not in the position where the normal vector is (1,1,1), but perhaps some other orientation. But according to the equations above, if the plane makes the same angle with all edges, then the normal vector must have equal components. Therefore, there's no other possibility. Therefore, the answer must be √3 / 3.Alternatively, let's think geometrically. The plane must be equally inclined to all three axes. The most symmetric such plane is the one with normal vector (1,1,1). Therefore, the angle calculation as above gives sin α = 1/√3.Therefore, after all these verifications, I can conclude that sin α = √3 / 3.Final AnswerThe value of sin alpha is boxed{dfrac{sqrt{3}}{3}}.</think>Given a plane that makes an angle (alpha) with all 12 edges of a cube, we need to find (sin alpha).1. Understanding the Cube and Plane: - A cube has 12 edges, each aligned with the x, y, or z-axis. - The plane must make the same angle (alpha) with all these edges.2. Angle Between a Line and a Plane: - The angle (alpha) between a line (edge) and a plane is the complement of the angle (theta) between the line and the normal vector of the plane. - (sin alpha = cos theta).3. Normal Vector of the Plane: - For the plane to be equally inclined to all edges, the normal vector must be equally inclined to all coordinate axes. - This implies the normal vector has equal components, such as ((1, 1, 1)).4. Calculating the Angle: - The normal vector ((1, 1, 1)) has magnitude (sqrt{1^2 + 1^2 + 1^2} = sqrt{3}). - The angle (theta) between the normal vector and any coordinate axis (edge) is (arccosleft(frac{1}{sqrt{3}}right)). - Therefore, (sin alpha = cos theta = frac{1}{sqrt{3}}).5. Simplifying the Result: - Rationalizing (frac{1}{sqrt{3}}) gives (frac{sqrt{3}}{3}).Thus, the value of (sin alpha) is (boxed{dfrac{sqrt{3}}{3}}).,<think>Alright, so I have this problem here: Given a plane that makes an angle α with all 12 edges of a cube, then sin α equals what? Hmm, okay. Let me think about how to approach this.First off, let me visualize a cube. A cube has 12 edges, right? Each edge is of equal length, and they all meet at right angles. Now, the problem says there's a plane that makes an angle α with all 12 edges. That seems interesting because a plane can't be parallel to all edges, and making the same angle with all edges seems like a special orientation. I need to figure out what that angle is and then find its sine.Wait, how does a plane make an angle with a line? Oh, right! The angle between a line and a plane is the complement of the angle between the line and the normal to the plane. So, if θ is the angle between the line and the normal, then α = 90° - θ. But maybe I should recall the exact definition here. Let me confirm.Yes, the angle between a line and a plane is defined as the angle between the line and its orthogonal projection onto the plane, which is indeed 90° minus the angle between the line and the normal vector of the plane. So, sin α would be sin(90° - θ) = cos θ. Therefore, sin α equals cos θ, where θ is the angle between the edges and the normal to the plane. Hmm, maybe this will be useful later.Since all 12 edges must make the same angle α with the plane, their projections onto the plane must be equal in some way. Let me consider the cube in a coordinate system. Let's suppose the cube is axis-aligned with edges along the x, y, and z-axes. Then the edges can be represented as vectors along the coordinate axes: (±1, 0, 0), (0, ±1, 0), (0, 0, ±1), right? Each edge is a vector of length 1 (assuming unit cube for simplicity).Now, the plane in question makes an angle α with each of these 12 edges. So, the angle between each edge vector and the plane is α. Let me denote the normal vector of the plane as n = (a, b, c). Since the plane makes the same angle with all edges, the angle between each edge vector and n should be the same θ (where θ = 90° - α). Therefore, the angle between n and each axis-aligned vector (1,0,0), (0,1,0), (0,0,1) should be equal.Let me recall that the angle φ between two vectors u and v is given by the formula cos φ = (u • v) / (|u| |v|). Here, n is (a, b, c) and the edge vectors are (1,0,0), etc. So, the dot product between n and (1,0,0) is a, and similarly b and c for the other axes. The magnitude of n is sqrt(a² + b² + c²), and each edge vector has magnitude 1.Therefore, cos θ = a / |n|, cos θ = b / |n|, and cos θ = c / |n|. Since all these are equal, that implies a = b = c. So, the normal vector must be along the line (a, a, a). Therefore, the normal vector is a scalar multiple of (1,1,1). Therefore, the plane in question is either x + y + z = d or some scalar multiple of that equation. Since the angle doesn't depend on d, we can assume d=0 for simplicity, so the plane is x + y + z = 0.Wait, but the cube is in a coordinate system where its edges are along the axes. So, the plane x + y + z = 0 is passing through the origin, but the cube might be placed such that its center is at the origin. But maybe since we're dealing with angles, the position of the plane (the d in the equation) doesn't matter, only the orientation. Therefore, the normal vector is (1,1,1), so the angle θ between n and each axis is equal. Let's compute cos θ.For example, take the x-axis edge vector (1,0,0). The dot product with n is 1*1 + 0*1 + 0*1 = 1. The magnitude of n is sqrt(1² + 1² + 1²) = sqrt(3). Therefore, cos θ = 1 / sqrt(3). Therefore, θ = arccos(1/√3). Then, since α = 90° - θ, sin α = sin(90° - θ) = cos θ = 1/√3. Wait, but that would mean sin α = 1/√3 ≈ 0.577. Is that the answer?Wait, but hold on. Let me confirm this again. If the angle between the edge and the plane is α, and the angle between the edge and the normal is θ, then α = 90° - θ. Therefore, sin α = sin(90° - θ) = cos θ. But cos θ here is 1/√3, so sin α = 1/√3. Therefore, is the answer √3 / 3? Because 1/√3 is equal to √3/3. Yes, rationalizing the denominator, that's correct.But let me check with another edge to confirm. Take the edge along the y-axis (0,1,0). The dot product with n is 1 as well. Same with the z-axis. So, all edges make the same angle θ with the normal, so α is the same for all edges. Therefore, sin α = 1/√3 = √3/3.But wait, is there another plane that could satisfy this condition? Because the problem states "a plane", not necessarily the one with normal (1,1,1). But in reality, the normal vector must be equally inclined to all three axes, which only happens when a = b = c. So, the only possible normals are scalar multiples of (1,1,1). Therefore, regardless of the scalar multiple, the direction is the same, so the angle θ remains arccos(1/√3). Therefore, sin α = √3/3.Alternatively, maybe I can think geometrically. If the plane is equally inclined to all edges, then it should be symmetric with respect to the cube's edges. The most symmetric plane relative to a cube is the one cutting through the midpoints of the edges or something. But perhaps the plane normal to the space diagonal (1,1,1). That plane is indeed symmetric with respect to the cube.Alternatively, let's consider projecting all edges onto the plane. If each edge makes the same angle with the plane, then the projection of each edge onto the plane would have the same length. The projection length of a vector onto a plane is |v| cos α, where α is the angle between the vector and the plane. Since all edges are of unit length, their projections would be cos α. Therefore, all edges project to the same length on the plane.But in order for all edges to project to the same length on the plane, the plane must be such that the angle between each edge and the plane is equal. So, that's again achieved when the normal vector is equally inclined to all three axes, which is the (1,1,1) direction.Alternatively, perhaps I can use coordinates. Let me set up a coordinate system where the cube has vertices at (0,0,0) to (1,1,1). Then the edges are along the coordinate axes. Let the plane have normal vector (a,b,c). The angle between the plane and each edge is α. As before, the angle between the edge and the normal is θ, and sin α = cos θ.So, for the edge along the x-axis, the angle θ between (1,0,0) and (a,b,c) is such that cos θ = a / sqrt(a² + b² + c²). Similarly for y and z axes: cos θ = b / sqrt(a² + b² + c²) and c / sqrt(a² + b² + c²). Since all these cos θ must be equal (since sin α must be the same for all edges), then a = b = c. Therefore, the normal vector must be (a,a,a), so direction (1,1,1).Therefore, cos θ = a / sqrt(3a²) = 1/√3. Therefore, sin α = cos θ = 1/√3, which is √3/3. Therefore, sin α = √3/3. So, the answer is √3/3. Therefore, boxed, it would be boxed{dfrac{sqrt{3}}{3}}.Wait, but let me just check again with an example. Let's take the normal vector as (1,1,1). Then the angle between (1,0,0) and (1,1,1) is θ, where cos θ = 1 / sqrt(3). Therefore, θ = arccos(1/√3). Then α = 90° - θ. Therefore, sin α = sin(90° - θ) = cos θ = 1/√3. Yes, exactly. So, sin α = √3/3.Alternatively, if I think in terms of direction ratios. If the plane is inclined at angle α to all edges, then for each edge, the angle between the edge and the plane is α, which as per definition, the angle between the line and the plane is the angle between the line and its projection onto the plane. So, if the original edge has length 1, then the projection onto the plane has length cos α. Therefore, all edges projected onto the plane have length cos α. For this to be true for all edges, regardless of their direction, the plane must be oriented such that each axis is equally "squashed" when projected onto the plane. Which again leads to the normal vector being along (1,1,1).Alternatively, maybe using linear algebra. For the plane to have equal angles with all edges, which are along the coordinate axes, the normal vector must have equal angles with each coordinate axis. Therefore, the direction cosines of the normal vector with each axis must be equal. The direction cosines are the cosines of the angles between the normal and each axis, which are a/|n|, b/|n|, c/|n|. For these to be equal, a = b = c. Therefore, the normal vector is (1,1,1), leading to the previous result.Thus, after considering multiple approaches—coordinate systems, geometric interpretations, direction cosines—it seems consistent that sin α is √3/3. So, I think that's the answer. But just to be thorough, let's imagine a specific example.Take a cube with edge length √3. Then the normal vector (1,1,1) has length √3. The angle θ between (1,0,0) and (1,1,1) is arccos( (1*1 + 0*1 + 0*1)/ (|(1,0,0)| * |(1,1,1)|) ) = arccos(1 / √3). Then α = 90° - θ. So, sin α = cos θ = 1/√3, which is √3/3. Yep, that checks out.Alternatively, if I use a unit cube, edge length 1. Then the normal vector (1,1,1) has length √3. The angle θ is still arccos(1/√3). Therefore, sin α = 1/√3 = √3/3. Same result.Therefore, regardless of cube size, the sine of the angle α is √3/3. Therefore, the answer should be boxed{dfrac{sqrt{3}}{3}}.Final Answerboxed{dfrac{sqrt{3}}{3}}</think>Given a plane that makes an angle (alpha) with all 12 edges of a cube, we need to find (sin alpha).First, we note that the angle (alpha) between a line (edge of the cube) and a plane is the complement of the angle (theta) between the line and the normal to the plane. Therefore, (alpha = 90^circ - theta), which implies (sin alpha = cos theta).We consider the cube in a coordinate system with edges along the x, y, and z-axes. The normal vector to the plane must make equal angles with all three coordinate axes. Let the normal vector be (mathbf{n} = (a, b, c)). The angle (theta) between (mathbf{n}) and each coordinate axis (x, y, z) is given by the direction cosines:[cos theta = frac{a}{|mathbf{n}|}, quad cos theta = frac{b}{|mathbf{n}|}, quad cos theta = frac{c}{|mathbf{n}|}]For these to be equal, we must have (a = b = c). Thus, the normal vector is a scalar multiple of ((1, 1, 1)). The magnitude of this normal vector is (sqrt{1^2 + 1^2 + 1^2} = sqrt{3}).The cosine of the angle (theta) between the normal vector and any edge (coordinate axis) is:[cos theta = frac{1}{sqrt{3}}]Since (sin alpha = cos theta), we have:[sin alpha = frac{1}{sqrt{3}} = frac{sqrt{3}}{3}]Thus, the final answer is (boxed{dfrac{sqrt{3}}{3}}).]

🔑:frac{sqrt{3}}{3}5.【Analysis and Solution】As shown in the figure, according to the problem, the plane BCD forms an angle alpha with all 12 edges of the cube. Draw AH perp plane BCD through point A, and connect DH, then alpha=angle ADH. Let the side length of the cube be a, then it is easy to find that DH=frac{2}{3} times sqrt{2} a cdot sin 60^{circ}begin{array}{c}=frac{sqrt{6}}{3} a, AH=sqrt{a^{2}-left(frac{sqrt{6}}{3} aright)^{2}}=frac{sqrt{3}}{3} a . therefore sin alpha=sin angle ADH=frac{AH}{AD}=frac{sqrt{3}}{3} .end{array}

🧠:[<think>Okay, so I have this system of equations to solve. Let me look at them again. The first equation is y equals 1 plus log base 4 of x, and the second equation is x raised to the power of y equals 4 to the power of 6. Hmm. I need to find the values of x and y that satisfy both equations. Alright, let's start by understanding each equation and see how they can be connected.First, the logarithmic equation: y = 1 + log₄x. That means y is defined in terms of x here. Maybe I can substitute this expression for y into the second equation. The second equation is x^y = 4^6. Since 4^6 is 4096, maybe simplifying that later. But first, let's see if substituting y from the first equation into the second equation works.So if y is 1 + log₄x, then x^y becomes x^(1 + log₄x). And that should equal 4^6. So the equation becomes x^(1 + log₄x) = 4^6. Now, how do I solve for x here? This seems a bit complicated. Maybe I can express x in terms of log base 4 first. Let's recall that log₄x is the exponent to which 4 must be raised to get x. So if I let log₄x = t, then x = 4^t. That substitution might make things easier.Let me try that. Let’s set t = log₄x. Then x = 4^t. Then the first equation becomes y = 1 + t. Now substitute both x and y into the second equation. The second equation is x^y = 4^6. Substituting x = 4^t and y = 1 + t, we get (4^t)^(1 + t) = 4^6.Simplify the left side. When you raise a power to a power, you multiply the exponents. So (4^t)^(1 + t) = 4^(t*(1 + t)). So that's 4^(t^2 + t) = 4^6. Since the bases are the same, we can set the exponents equal. Therefore, t^2 + t = 6.Now we have a quadratic equation: t² + t - 6 = 0. Let me solve this quadratic. The quadratic formula is t = [-b ± sqrt(b² - 4ac)]/(2a). Here, a = 1, b = 1, c = -6. Plugging in, we get t = [-1 ± sqrt(1 + 24)]/2 = [-1 ± sqrt(25)]/2 = [-1 ± 5]/2. So two solutions: t = (-1 + 5)/2 = 4/2 = 2, and t = (-1 -5)/2 = -6/2 = -3. So t = 2 or t = -3.But t is log₄x. So let's find x for each t. If t = 2, then x = 4^2 = 16. If t = -3, then x = 4^(-3) = 1/(4^3) = 1/64. Then, using the first equation, y = 1 + t. So for t = 2, y = 1 + 2 = 3. For t = -3, y = 1 + (-3) = -2. So the solutions are (x, y) = (16, 3) and (1/64, -2). But wait, I need to check if these solutions satisfy both original equations, just to be sure.First, let's check (16, 3). First equation: y = 1 + log₄x. log₄16 is 2 because 4^2 = 16. So 1 + 2 = 3, which matches y = 3. Second equation: x^y = 16^3. Let's calculate 16^3. 16*16=256, 256*16=4096. And 4^6 is also 4096, so that's good. Now check the other solution, (1/64, -2). First equation: log₄(1/64). Since 1/64 is 4^(-3), log base 4 of 4^(-3) is -3. So 1 + (-3) = -2, which matches y = -2. Second equation: x^y = (1/64)^(-2). A negative exponent means reciprocal, so (1/64)^(-2) = (64)^2. 64 squared is 4096, which again is 4^6. So both solutions check out.Wait a second, but logarithms are only defined for positive real numbers. So x has to be positive. In both solutions, x is positive (16 and 1/64), so that's okay. Also, when we took log₄x, x is positive, which is valid. So both solutions are valid. Therefore, the system has two solutions: (16, 3) and (1/64, -2). Hmm, but let me make sure I didn't make any mistakes in substitution.Wait, when we set t = log₄x, and then x = 4^t, that's correct because 4^(log₄x) = x. Then substituting into x^y gives (4^t)^(1 + t) = 4^(t*(1 + t)) = 4^6. So exponent equations. Then solving t² + t - 6 = 0, which factors? Let me check. t² + t -6. Let's see if it factors. (t + 3)(t - 2) = t² + t -6. Yes, exactly. So roots are t = -3 and t = 2. So that's correct. Then x = 4^t gives x = 4^(-3) = 1/64 and x = 16. Then y is 1 + t, so y = -2 and 3. So all steps seem correct.Alternatively, another approach could be to express everything in terms of base 4. Let's try that. Starting with the original equations.First equation: y = 1 + log₄x. Let's rewrite log₄x as (ln x)/(ln 4), but maybe not necessary. Alternatively, since the second equation is x^y = 4^6, take log base 4 of both sides. So log₄(x^y) = log₄(4^6). Which simplifies to y * log₄x = 6. So y * log₄x = 6. But from the first equation, log₄x = y -1. So substituting that into the second equation: y*(y -1) = 6. So y² - y - 6 = 0. Solving this quadratic equation: y² - y -6 = 0. Using quadratic formula, y = [1 ± sqrt(1 + 24)]/2 = [1 ± 5]/2. So y = (1 + 5)/2 = 3, and y = (1 -5)/2 = -2. Then, log₄x = y -1. For y=3, log₄x = 3 -1 = 2, so x = 4² =16. For y=-2, log₄x = -2 -1 = -3, so x =4^(-3)=1/64. So same solutions. This method is quicker. So either way, same solutions.Therefore, the solutions are (16,3) and (1/64, -2). So I think that's correct. Both methods lead to the same answers, and both solutions satisfy the original equations. So I don't see any mistakes here. Let me just recap.First method: substitute y from the first equation into the second equation, leading to an equation in x. Then use substitution t = log₄x, leading to quadratic in t. Solve for t, then get x and y.Second method: take log base 4 of both sides in the second equation, then substitute log₄x from the first equation, leading to quadratic in y. Solve for y, then get x.Both approaches are valid and lead to the same solutions. So yeah, the solutions are (16,3) and (1/64, -2). Since the problem is presented as a system, and there are two solutions, both should be included.Wait, but sometimes, depending on the context, even if algebraically there are two solutions, sometimes in the original equations, there might be restrictions. For example, if x has to be positive, which it is here. Or maybe y has to be an integer or something? But the problem doesn't specify any restrictions. So both solutions are acceptable.Therefore, the final answer is x = 16 and y = 3, or x = 1/64 and y = -2.Final AnswerThe solutions are (boxed{(16, 3)}) and (boxed{left( dfrac{1}{64}, -2 right)}).</think>Given the system of equations:[left{begin{array}{l}y = 1 + log_{4} x, x^{y} = 4^{6}end{array}right.]We start by substituting ( y = 1 + log_{4} x ) from the first equation into the second equation. This gives:[x^{1 + log_{4} x} = 4^{6}]Let ( t = log_{4} x ), which implies ( x = 4^{t} ). Substituting ( x = 4^{t} ) and ( y = 1 + t ) into the second equation, we get:[(4^{t})^{1 + t} = 4^{6}]Simplifying the left side, we have:[4^{t(t + 1)} = 4^{6}]Equating the exponents, we get the quadratic equation:[t^2 + t - 6 = 0]Solving this quadratic equation using the quadratic formula ( t = frac{-b pm sqrt{b^2 - 4ac}}{2a} ) with ( a = 1 ), ( b = 1 ), and ( c = -6 ):[t = frac{-1 pm sqrt{1 + 24}}{2} = frac{-1 pm 5}{2}]This gives the solutions:[t = 2 quad text{and} quad t = -3]For ( t = 2 ):- ( x = 4^2 = 16 )- ( y = 1 + 2 = 3 )For ( t = -3 ):- ( x = 4^{-3} = frac{1}{64} )- ( y = 1 + (-3) = -2 )Both solutions are checked and satisfy the original equations. Therefore, the solutions to the system are:[boxed{(16, 3)} quad text{and} quad boxed{left( dfrac{1}{64}, -2 right)}],<think>Okay, so I need to solve this system of equations. Let me see. The first equation is y equals 1 plus log base 4 of x. The second equation is x raised to the power of y equals 4 to the sixth power. Hmm. Alright, let's break this down step by step.First, since the first equation gives y in terms of x, maybe I can substitute that into the second equation. That way, I can have an equation with just x and solve for it. Let me try that.So, the first equation is y = 1 + log₄x. The second equation is x^y = 4⁶. If I substitute the expression for y from the first equation into the second equation, I'll get x^(1 + log₄x) = 4⁶. Hmm, that seems manageable. Let me write that out:x^(1 + log₄x) = 4⁶.Now, I need to solve for x. Let's see. Maybe I can express x in terms of base 4 since the logarithm is base 4. Let me recall that log₄x is the exponent to which we raise 4 to get x. So, maybe I can write x as 4^k, where k is some number. Let me try that substitution.Let’s set x = 4^k. Then, log₄x would be log₄(4^k) which is k, because logarithm base 4 of 4^k is just k. So, substituting x = 4^k into the first equation, we get y = 1 + k.Now, substitute x = 4^k and y = 1 + k into the second equation, which is x^y = 4⁶. Let's do that:(4^k)^(1 + k) = 4⁶.Simplify the left side. When you raise a power to a power, you multiply the exponents. So, 4^(k*(1 + k)) = 4⁶.Since the bases are the same and the equation holds, the exponents must be equal. Therefore:k*(1 + k) = 6.Let me write that as a quadratic equation:k² + k - 6 = 0.Now, solving for k. Let's use the quadratic formula. For ax² + bx + c = 0, the solutions are (-b ± sqrt(b² - 4ac)) / 2a.Here, a = 1, b = 1, c = -6. Plugging in:k = [-1 ± sqrt(1² - 4*1*(-6))]/(2*1)= [-1 ± sqrt(1 + 24)]/2= [-1 ± sqrt(25)]/2= [-1 ± 5]/2.So, two solutions:k = (-1 + 5)/2 = 4/2 = 2,k = (-1 - 5)/2 = -6/2 = -3.Therefore, k = 2 or k = -3.Now, since x = 4^k, let's find the corresponding x values.First, for k = 2:x = 4² = 16.Then, y = 1 + k = 1 + 2 = 3.Second, for k = -3:x = 4^(-3) = 1/(4³) = 1/64.Then, y = 1 + (-3) = -2.So, we have two solutions: (x = 16, y = 3) and (x = 1/64, y = -2).But wait, we should check if these solutions satisfy the original equations. Let me verify both solutions in both equations.First solution: x = 16, y = 3.First equation: y = 1 + log₄x. Let's check: log₄16. Since 4² = 16, log₄16 = 2. So, 1 + 2 = 3, which matches y = 3. Good.Second equation: x^y = 16³. Let's compute 16³. 16*16=256, 256*16=4096. 4⁶ is 4096 as well (since 4⁵=1024, 4⁶=4096). So, 16³=4096=4⁶. Correct.Second solution: x = 1/64, y = -2.First equation: y = 1 + log₄(1/64). Let's compute log₄(1/64). 1/64 is 4^(-3), since 4³=64, so reciprocal is 4^(-3). Therefore, log₄(1/64) = -3. So, 1 + (-3) = -2, which matches y = -2. Good.Second equation: x^y = (1/64)^(-2). Let's compute that. (1/64)^(-2) = (64)^2. 64 squared is 4096. And 4⁶ is 4096. So, (1/64)^(-2) = 4096 = 4⁶. Correct.Therefore, both solutions are valid.But wait, hold on. In the original equations, is there any restriction on x or y? Let's check.For the logarithm log₄x to be defined, x must be positive. Since x is in the logarithm in the first equation, x > 0. Both solutions x = 16 and x = 1/64 are positive, so that's okay.For the second equation x^y = 4⁶, since x is positive, x^y is defined for any real y, so no problem there.So, both solutions are valid. Therefore, the system has two solutions: (16, 3) and (1/64, -2).But wait, let me check once more if there could be any mistakes in substitution or exponent rules.First solution: x=16, y=3. Plug into second equation: 16^3=4096, and 4^6=4096. Correct.Second solution: x=1/64, y=-2. Plug into second equation: (1/64)^-2 = 64^2 = 4096, which equals 4^6. Correct.So both solutions check out. Therefore, the answer should be both pairs.But let me think again: when dealing with logarithms, sometimes extraneous solutions can appear, but in this case, since we did substitution and solved step by step without squaring or anything that might introduce extraneous roots, both solutions are valid. So, the system has two solutions.Therefore, the solutions are x = 16, y = 3 and x = 1/64, y = -2.Final AnswerThe solutions are (boxed{(16, 3)}) and (boxed{left( dfrac{1}{64}, -2 right)}).</think>Given the system of equations:[left{begin{array}{l}y=1+log _{4} x, x^{y}=4^{6}end{array}right.]First, substitute ( y = 1 + log_{4} x ) from the first equation into the second equation:[x^{1 + log_{4} x} = 4^{6}]Express ( x ) as ( 4^k ). Then, ( log_{4} x = k ), so ( y = 1 + k ). Substituting ( x = 4^k ) and ( y = 1 + k ) into the second equation:[(4^k)^{1 + k} = 4^{6}]Simplify the left side:[4^{k(1 + k)} = 4^{6}]Equating the exponents:[k(1 + k) = 6]Solving the quadratic equation ( k^2 + k - 6 = 0 ) using the quadratic formula:[k = frac{-1 pm sqrt{1 + 24}}{2} = frac{-1 pm 5}{2}]This gives the solutions:[k = 2 quad text{or} quad k = -3]For ( k = 2 ):- ( x = 4^2 = 16 )- ( y = 1 + 2 = 3 )For ( k = -3 ):- ( x = 4^{-3} = frac{1}{64} )- ( y = 1 - 3 = -2 )Both solutions are verified to satisfy the original equations. Therefore, the solutions are:[boxed{(16, 3)} quad text{and} quad boxed{left( dfrac{1}{64}, -2 right)}]]

🔑:## Solution.Domain of definition: 0<x neq 1.Taking the logarithm of the second equation of the system with base 4, we have log _{4} x^{y}=log _{4} 4^{6}, y log _{4} x=6. Therefore,left{begin{array}{l}y=1+log _{4} x, y log _{4} x=6end{array} Rightarrowleft(1+log _{4} xright) log _{4} x=6, log _{4}^{2} x+log _{4} x-6=0right.Solving this equation as a quadratic equation in terms of log _{4} x, we find left(log _{4} xright)_{1}=-3,left(log _{4} xright)_{2}=2, x_{1}=frac{1}{64}, x_{2}=16. Then y_{1}=-2, y_{2}=3.Answer: left(frac{1}{64} ;-2right),(16 ; 3).

🧠:[<think>Okay, so there's this problem here: 6 athletes participate in the final, and there are no ties for any positions. We need to find the number of possible outcomes where Athlete No. 1 does not come in sixth place. Hmm, let me think through this step by step.First, let me make sure I understand the problem correctly. There are 6 athletes, each finishing in a unique position from 1st to 6th. So, the total number of possible outcomes without any restrictions would just be the number of permutations of 6 athletes, right? Which is 6 factorial, 6! So 6×5×4×3×2×1 = 720. That makes sense.But the question is asking for the number of outcomes where Athlete No. 1 does NOT come in sixth place. So, I need to subtract the number of outcomes where Athlete No. 1 is in sixth place from the total number of outcomes. That seems like a standard approach for these "at least one" or "not" problems—use complementary counting.So, total outcomes: 6! = 720.Now, number of outcomes where Athlete No. 1 is in sixth place. Let's figure that out. If Athlete No. 1 is fixed in sixth place, then the remaining 5 athletes can finish in any of the remaining positions (1st to 5th). So, the number of such outcomes is 5! = 120.Therefore, the number of outcomes where Athlete No. 1 is not in sixth place would be total outcomes minus restricted outcomes: 720 - 120 = 600.Wait, that seems straightforward, but let me double-check to make sure I didn't miss anything. So, the problem is only restricting Athlete No. 1 from being in sixth place. There are no other restrictions on the other athletes or on Athlete No. 1's position otherwise. So, yes, it's correct that we just subtract the cases where Athlete No. 1 is in sixth place.Alternatively, another way to think about it is: How many possible positions can Athlete No. 1 take? Since they can't be in sixth place, they have 5 possible positions (1st to 5th). Once Athlete No. 1's position is fixed, the remaining 5 athletes can be arranged in the remaining 5 positions. So, that would be 5 × 5! = 5 × 120 = 600. Which matches the previous result. So that's a good consistency check.Therefore, both methods give the same answer of 600. Let me just confirm once more. Total permutations are 720. If Athlete No. 1 is in sixth place, the rest can be arranged in 5! ways, which is 120. 720 - 120 = 600. Yep, that's solid.Alternatively, using permutations where one element is restricted: For any permutation problem where a specific element can't be in a specific position, the formula is (n-1) × (n-1)! if only one position is restricted? Wait, no, that's maybe a different problem. Wait, if we have n elements and one element can't be in a specific position, how do we compute that?Wait, for example, derangements are when elements can't be in certain positions, but here it's a single element and a single position. So, for each case, the number of permutations where element x is not in position k is total permutations minus permutations where x is in position k. Which is n! - (n-1)! So, 6! - 5! = 720 - 120 = 600. Which is exactly what we have here. So, that formula holds here. So, the answer is 720 - 120 = 600.Alternatively, if you think of it as: How many choices for the sixth position? Since Athlete No. 1 can't be there, there are 5 choices (Athletes 2-6). Once the sixth position is filled by someone else, the remaining 5 athletes (including Athlete No. 1) can be arranged in the first five positions. So that's 5 × 5! = 5 × 120 = 600. Yep, same answer. So, multiple approaches confirm the result.I don't see any flaws in this reasoning. The key steps are recognizing that the total number of possible outcomes is 6!, the number of outcomes where Athlete 1 is in 6th place is 5!, subtracting them gives the desired count. Alternatively, assigning the restricted position first and then permuting the rest. Both methods work, and both give 600. So, I think that's the answer.Final AnswerThe number of possible outcomes is boxed{600}.</think>To find the number of possible outcomes where Athlete No. 1 does not come in sixth place, we start by calculating the total number of permutations of the 6 athletes, which is (6!):[6! = 720]Next, we calculate the number of outcomes where Athlete No. 1 is in sixth place. If Athlete No. 1 is fixed in sixth place, the remaining 5 athletes can be arranged in any order in the remaining positions. This is given by (5!):[5! = 120]Subtracting the number of restricted outcomes from the total outcomes gives us the number of valid outcomes where Athlete No. 1 is not in sixth place:[720 - 120 = 600]Thus, the number of possible outcomes where Athlete No. 1 does not come in sixth place is (boxed{600}).,<think>Okay, let's see. The problem is about 6 athletes participating in a final, and we need to find the number of possible outcomes where Athlete No. 1 doesn't come in sixth place. Hmm, no ties allowed, so each position from first to sixth is occupied by a different athlete.First, I remember that when there are no restrictions, the number of possible outcomes for the race is just the number of permutations of 6 athletes. Since each position is distinct and no ties are allowed, that's 6 factorial, right? So 6! = 720. That's the total number of ways the athletes can finish.But the problem wants the number of outcomes where Athlete No. 1 is not in sixth place. So instead of calculating this directly, maybe it's easier to calculate the total number of outcomes and subtract the number of outcomes where Athlete No. 1 is in sixth place. That should give the desired number. Yeah, that makes sense. Sometimes subtracting the unwanted cases from the total is simpler.Let me verify. Total outcomes: 6! = 720. Now, how many outcomes have Athlete No. 1 in sixth place? If we fix Athlete No. 1 in sixth position, then the remaining five athletes can finish in the first five positions. So that's 5! permutations for the other athletes. 5! is 120. So, the number of outcomes where Athlete No. 1 is in sixth place is 120. Therefore, the number of outcomes where he is not in sixth place should be 720 - 120 = 600.Wait, but let me think again. Another way to approach it is to consider the possible positions for Athlete No. 1. He can't be in sixth place, so he has 5 possible positions (first to fifth). For each of these positions, the remaining five athletes can be arranged in the remaining five positions. So that would be 5 (choices for Athlete No. 1's position) multiplied by 5! (arrangements of the others). Let's compute that: 5 * 120 = 600. Yep, same result. So both methods give 600. That seems consistent.Is there another way to check this? Maybe by considering probability. The probability that Athlete No. 1 is in sixth place is 1/6, since each position is equally likely assuming all permutations are equally probable. Therefore, the number of favorable outcomes is total outcomes multiplied by (1 - 1/6) = 5/6. So 720 * (5/6) = 600. That matches too. So three different approaches all lead to 600. That gives me more confidence that the answer is correct.I think that's thorough. So the answer should be 600. Let me just recap: total permutations minus permutations where Athlete 1 is sixth, which is 720 - 120 = 600. Yep, that's straightforward.Final AnswerThe number of possible outcomes is boxed{600}.</think>To find the number of possible outcomes where Athlete No. 1 does not come in sixth place, we start by calculating the total number of permutations of 6 athletes, which is (6!):[6! = 720]Next, we calculate the number of permutations where Athlete No. 1 is in sixth place. If Athlete No. 1 is fixed in sixth position, the remaining 5 athletes can be arranged in the first five positions in (5!) ways:[5! = 120]Subtracting the number of unwanted outcomes from the total outcomes gives us the number of favorable outcomes:[720 - 120 = 600]We can also verify this by considering the possible positions for Athlete No. 1. Since Athlete No. 1 cannot be in sixth place, there are 5 possible positions for him. For each of these positions, the remaining 5 athletes can be arranged in (5!) ways:[5 times 5! = 5 times 120 = 600]Another verification using probability: the probability that Athlete No. 1 is not in sixth place is (1 - frac{1}{6} = frac{5}{6}). Multiplying this probability by the total number of outcomes:[720 times frac{5}{6} = 600]Thus, the number of possible outcomes where Athlete No. 1 does not come in sixth place is (boxed{600}).]

🔑:Solving: Since there are no parallel rankings, the number of ranking situations for 6 athletes is 6!. If athlete No.1 gets the last place, the number of situations for the remaining 5 athletes to get the top five places is 5!. Therefore, by the method of exclusion, the number of situations where athlete No.1 does not get the last place is 6!-5!=5 cdot 5!=600 kinds.

🧠:[<think>Okay, let's tackle this problem step by step. The question is about finding the locus of complex numbers z that satisfy the equation 2|z - 2 + i| + 2|z + 3 - i| = 12. The options are Circle, Ellipse, Hyperbola, or Parabola. Hmm, complex numbers and loci... I remember that complex numbers can be represented as points in the plane, so z corresponds to a point (x, y) where z = x + yi. First, let's rewrite the equation in terms of x and y. Let me substitute z with x + yi. So, z - 2 + i becomes (x - 2) + (y + 1)i, and z + 3 - i becomes (x + 3) + (y - 1)i. The modulus of a complex number a + bi is sqrt(a² + b²), so these modulus expressions represent distances. Therefore, |z - 2 + i| is the distance from the point (x, y) to (2, -1), since the real part is x - 2 and the imaginary part is y + 1. Similarly, |z + 3 - i| is the distance from (x, y) to (-3, 1). So the equation becomes 2 times the distance from (x, y) to (2, -1) plus 2 times the distance from (x, y) to (-3, 1) equals 12. Let me factor out the 2. That gives 2[ |z - 2 + i| + |z + 3 - i| ] = 12. Dividing both sides by 2, we get |z - 2 + i| + |z + 3 - i| = 6. So now the equation is the sum of the distances from (x, y) to the two fixed points (2, -1) and (-3, 1) equals 6. Wait, that sounds familiar. In analytic geometry, the set of points where the sum of the distances to two fixed points (foci) is constant is an ellipse, right? So if the sum is constant and greater than the distance between the foci, it's an ellipse. If it's equal, it's the line segment between them, and if it's less, then no such points exist. So first, let's check the distance between the two foci. The foci are (2, -1) and (-3, 1). The distance between them can be calculated using the distance formula: sqrt[(2 - (-3))² + (-1 - 1)²] = sqrt[(5)² + (-2)²] = sqrt[25 + 4] = sqrt[29]. Which is approximately 5.385. The constant sum given in the problem is 6. Since 6 is greater than sqrt(29) (~5.385), the locus should indeed be an ellipse. Therefore, the answer should be B. Ellipse. But wait, let me double-check. The original equation had 2 multiplied by each modulus term, adding up to 12. After dividing by 2, the sum of the distances is 6. So the sum is 6, which is greater than the distance between the two points. So yes, that's the definition of an ellipse. Alternatively, if it were a difference of distances, that would be a hyperbola. But here it's a sum. So I think it's definitely an ellipse. So the answer is B.Final Answerboxed{B}</think>The problem requires determining the locus of complex numbers ( z ) that satisfy the equation ( 2|z - 2 + i| + 2|z + 3 - i| = 12 ).1. Convert the complex numbers to points in the plane: - ( z = x + yi ) corresponds to the point ( (x, y) ). - ( |z - 2 + i| ) represents the distance from ( (x, y) ) to ( (2, -1) ). - ( |z + 3 - i| ) represents the distance from ( (x, y) ) to ( (-3, 1) ).2. Simplify the given equation: [ 2|z - 2 + i| + 2|z + 3 - i| = 12 ] Dividing both sides by 2: [ |z - 2 + i| + |z + 3 - i| = 6 ] This equation states that the sum of the distances from the point ( (x, y) ) to the two fixed points ( (2, -1) ) and ( (-3, 1) ) is 6.3. Check the distance between the two fixed points: - The distance between ( (2, -1) ) and ( (-3, 1) ) is calculated using the distance formula: [ sqrt{(2 - (-3))^2 + (-1 - 1)^2} = sqrt{5^2 + (-2)^2} = sqrt{25 + 4} = sqrt{29} approx 5.385 ] - Since the sum of the distances (6) is greater than the distance between the points (( sqrt{29} approx 5.385 )), the locus is an ellipse.Thus, the answer is (boxed{B}).,<think>Okay, so I need to figure out the locus of complex numbers z that satisfy the equation 2|z - 2 + i| + 2|z + 3 - i| = 12. The options are Circle, Ellipse, Hyperbola, or Parabola. Hmm. Let me start by recalling what these terms mean in the context of complex numbers. First, complex numbers can be represented as points in the plane, where z = x + yi corresponds to the point (x, y). So, the equation given is in terms of the modulus (or absolute value) of complex numbers, which translates to distances between points in the plane. The equation is 2|z - (2 - i)| + 2|z - (-3 + i)| = 12. Let me rewrite that for clarity. Let me denote the two complex numbers as points A and B. Point A is (2, -1) because when z = 2 - i, the real part is 2 and the imaginary part is -1 (since i has a coefficient of -1). Similarly, point B is (-3, 1) because z = -3 + i corresponds to (-3, 1). So the equation becomes 2 times the distance from z to A plus 2 times the distance from z to B equals 12. If I divide both sides by 2, I get |z - A| + |z - B| = 6. Wait a second, that equation |z - A| + |z - B| = constant is familiar. Isn't that the definition of an ellipse? Because an ellipse is the set of all points where the sum of the distances to two fixed points (the foci) is constant. So, if that's the case, then this should be an ellipse. But let me verify if the constant here is greater than the distance between A and B, which is necessary for it to be an ellipse.First, let's compute the distance between points A and B. Point A is (2, -1) and point B is (-3, 1). Using the distance formula: sqrt[(2 - (-3))² + (-1 - 1)²] = sqrt[(5)² + (-2)²] = sqrt[25 + 4] = sqrt[29]. So the distance between A and B is sqrt(29). The constant in the equation is 6. Let's check if 6 > sqrt(29). Well, sqrt(25) is 5, sqrt(29) is approximately 5.385, so 6 is indeed greater than sqrt(29). Therefore, this satisfies the condition for an ellipse, where the sum of the distances to the foci is greater than the distance between the foci. Therefore, the locus should be an ellipse. So the answer should be B. Ellipse. But wait, let me make sure I didn't misinterpret the original equation. The original equation was 2|z - 2 + i| + 2|z + 3 - i| = 12. So when I factor out the 2, it becomes 2(|z - (2 - i)| + |z - (-3 + i)|) = 12, which simplifies to |z - (2 - i)| + |z - (-3 + i)| = 6. So yes, that's exactly the sum of distances being 6. Therefore, since 6 is greater than the distance between the two points (approx 5.385), the locus is an ellipse.Alternatively, if the sum was equal to the distance between the points, it would be a line segment between them. If it was less, it would be the empty set. So yes, in this case, since 6 > sqrt(29), it's an ellipse. Therefore, the correct answer is B. Ellipse.Final Answerboxed{B}</think>The problem requires determining the locus of complex numbers ( z ) that satisfy the equation ( 2|z - 2 + i| + 2|z + 3 - i| = 12 ).First, we rewrite the equation by dividing both sides by 2:[|z - (2 - i)| + |z - (-3 + i)| = 6]This equation represents the sum of the distances from the complex number ( z ) to two fixed points ( A(2, -1) ) and ( B(-3, 1) ) being constant (6). Next, we check if this constant is greater than the distance between the two points ( A ) and ( B ). Using the distance formula:[text{Distance between } A text{ and } B = sqrt{(2 - (-3))^2 + (-1 - 1)^2} = sqrt{(5)^2 + (-2)^2} = sqrt{25 + 4} = sqrt{29} approx 5.385]Since 6 (the constant sum) is greater than ( sqrt{29} ) (the distance between ( A ) and ( B )), the locus is an ellipse. Thus, the correct answer is (boxed{B}).]

🔑:2. mathrm{B}|z-(2-mathrm{i})|+|z-(-3+mathrm{i})|=6