answer：First, I need to find all the positive factors of 24. To do this, I'll start by identifying the prime factors of 24.24 can be divided by 2: 24 ÷ 2 = 1212 can be divided by 2: 12 ÷ 2 = 66 can be divided by 2: 6 ÷ 2 = 33 can be divided by 3: 3 ÷ 3 = 1So, the prime factorization of 24 is 2³ × 3¹.Next, I'll use the formula for the sum of divisors. For a number expressed as a product of primes raised to their respective exponents, the sum of its positive divisors is the product of the sums of each prime's powers from 0 up to its exponent.Applying this to 24:Sum of divisors = (2⁰ + 2¹ + 2² + 2³) × (3⁰ + 3¹) = (1 + 2 + 4 + 8) × (1 + 3) = 15 × 4 = 60Therefore, the sum of the positive factors of 24 is 60.

answer：Okay, so I have this problem here about a factory producing three models of products: A, B, and C. The ratio of their quantities is 3:4:7. They’re using stratified sampling to draw a sample of size n, and in this sample, there are 15 pieces of product A. I need to find the sample size n.Hmm, stratified sampling... I remember that stratified sampling is a method where the population is divided into subgroups, or strata, and samples are taken from each stratum in proportion to their size in the population. So, in this case, the three products A, B, and C are the strata, and their quantities are in the ratio 3:4:7.First, let me figure out what the total ratio is. If the ratio is 3:4:7, then the total parts are 3 + 4 + 7. Let me calculate that: 3 + 4 is 7, and 7 + 7 is 14. So, the total ratio is 14 parts.Now, in stratified sampling, the sample size from each stratum is proportional to the size of the stratum in the population. That means the number of samples from each product should be in the same ratio as their quantities.Given that, the number of product A in the sample is 15. Since the ratio for A is 3, the proportion of A in the sample should be 3/14 of the total sample size n. So, mathematically, that would be:Number of A in sample = (3/14) * nWe know that the number of A in the sample is 15, so:15 = (3/14) * nTo find n, I can rearrange this equation. Let me do that step by step.First, write the equation:15 = (3/14) * nI need to solve for n. So, I can multiply both sides by the reciprocal of 3/14, which is 14/3.n = 15 * (14/3)Let me compute that. 15 divided by 3 is 5, and 5 multiplied by 14 is 70.So, n = 70.Wait, let me double-check that. If the ratio is 3:4:7, then the proportion for A is 3/14. If 3/14 of n is 15, then n is 15 * (14/3). 15 divided by 3 is 5, times 14 is 70. Yeah, that seems right.Alternatively, I can think about it in terms of fractions. If 3 parts correspond to 15, then each part corresponds to 15 / 3 = 5. So, each part is 5. Then, the total number of parts is 14, so n = 14 * 5 = 70. That also gives me 70.So, both methods give me the same answer, which is good. It seems consistent.Just to make sure, let me think about the other products as well. If the sample size is 70, then the number of product B should be (4/14)*70 = 20, and the number of product C should be (7/14)*70 = 35. So, 15 + 20 + 35 = 70, which checks out.Therefore, I think the sample size n is 70.**Final Answer**The sample size ( n ) is boxed{70}.

answer：First, I need to determine the total age of the original 20 teachers. Since the average age was 49 years, the total age is 20 multiplied by 49, which equals 980 years.When a new teacher joins, the total number of teachers becomes 21. The new average age is 48 years, so the total age for all 21 teachers is 21 multiplied by 48, resulting in 1008 years.To find the age of the new teacher, I subtract the original total age from the new total age: 1008 years minus 980 years equals 28 years.Therefore, the new teacher is 28 years old.

answer：Okay, so I have this sequence {a_n}, where each term is given by a_n = 1 divided by n times (n + 2). And I need to find out which term in this sequence is equal to 1/120. Hmm, let me think about how to approach this.First, let me write down the formula again to make sure I have it right: a_n = 1 / [n(n + 2)]. So each term is the reciprocal of n multiplied by (n + 2). I need to find the value of n such that a_n equals 1/120.So, setting up the equation: 1 / [n(n + 2)] = 1/120. I need to solve for n here. Since both sides are fractions, maybe I can cross-multiply to get rid of the denominators. Let me try that.Cross-multiplying gives me n(n + 2) = 120. So, expanding the left side, that's n squared plus 2n equals 120. Hmm, okay, so I have a quadratic equation here: n² + 2n - 120 = 0.Now, I need to solve this quadratic equation for n. Quadratic equations can be solved using factoring, completing the square, or the quadratic formula. Let me see if this one factors nicely.Looking for two numbers that multiply to -120 and add up to 2. Let me think... 12 and -10? Because 12 times -10 is -120, and 12 + (-10) is 2. Yeah, that works. So, I can factor the quadratic as (n + 12)(n - 10) = 0.Setting each factor equal to zero gives me n + 12 = 0 or n - 10 = 0. So, n = -12 or n = 10. But wait, n is a positive integer because it's in the set of natural numbers, right? So, n can't be negative. Therefore, n = 10 is the solution.Let me double-check my work to make sure I didn't make a mistake. Plugging n = 10 back into the original formula: a_10 = 1 / [10(10 + 2)] = 1 / (10 * 12) = 1 / 120. Yep, that's correct.Just to be thorough, let me make sure that n = -12 doesn't somehow work, even though it's negative. Plugging n = -12 into the formula: a_{-12} = 1 / [-12(-12 + 2)] = 1 / [-12 * (-10)] = 1 / 120. Hmm, that also gives 1/120. But since n has to be a positive integer, n = -12 isn't a valid term in the sequence. So, n = 10 is definitely the answer.I think that's solid. I don't see any mistakes in my reasoning. I set up the equation correctly, solved the quadratic, checked the solutions, and confirmed that only n = 10 is valid in this context. So, yeah, 1/120 is the 10th term of the sequence.**Final Answer**The 10th term.boxed{10}

answer：Okay, so I have this problem about a geometric sequence. Let me try to figure it out step by step. First, let's understand the problem.We have a geometric sequence {a_n} with a common ratio q = 2. That means each term is twice the previous one. So, if the first term is a_1, then the second term a_2 is 2*a_1, the third term a_3 is 2*a_2 = 4*a_1, and so on. So, in general, a_n = a_1 * 2^{n-1}.Now, the problem says that a_2, a_3 + 1, a_4 form an arithmetic sequence. Hmm, okay. So, an arithmetic sequence has a common difference between consecutive terms. That means the difference between the second and first term is the same as the difference between the third and second term.Let me write down what I know:- a_2 = 2*a_1- a_3 = 4*a_1- a_4 = 8*a_1So, the three terms in question are:- First term: a_2 = 2*a_1- Second term: a_3 + 1 = 4*a_1 + 1- Third term: a_4 = 8*a_1Since these form an arithmetic sequence, the difference between the second and first term should be equal to the difference between the third and second term. So, let's write that equation.The difference between the second and first term is (a_3 + 1) - a_2 = (4*a_1 + 1) - 2*a_1 = 2*a_1 + 1.The difference between the third and second term is a_4 - (a_3 + 1) = 8*a_1 - (4*a_1 + 1) = 4*a_1 - 1.Since these differences are equal, we have:2*a_1 + 1 = 4*a_1 - 1Let me solve this equation for a_1.Subtract 2*a_1 from both sides:1 = 2*a_1 - 1Then, add 1 to both sides:2 = 2*a_1Divide both sides by 2:a_1 = 1Okay, so the first term a_1 is 1. That makes sense. Now, let's find a_n.Since it's a geometric sequence with common ratio q = 2, the nth term is a_n = a_1 * 2^{n-1} = 1 * 2^{n-1} = 2^{n-1}.So, part (1) is done: a_1 = 1 and a_n = 2^{n-1}.Now, moving on to part (2). We have b_n = a_n + n. We need to calculate the sum of the first 5 terms of {b_n}, which is S_5.First, let's write down what b_n is. Since a_n = 2^{n-1}, then b_n = 2^{n-1} + n.So, the first 5 terms of {b_n} are:- b_1 = 2^{0} + 1 = 1 + 1 = 2- b_2 = 2^{1} + 2 = 2 + 2 = 4- b_3 = 2^{2} + 3 = 4 + 3 = 7- b_4 = 2^{3} + 4 = 8 + 4 = 12- b_5 = 2^{4} + 5 = 16 + 5 = 21Now, let's add these up:S_5 = b_1 + b_2 + b_3 + b_4 + b_5 = 2 + 4 + 7 + 12 + 21Let me compute this step by step:2 + 4 = 66 + 7 = 1313 + 12 = 2525 + 21 = 46So, S_5 = 46.Alternatively, maybe I can compute it using a formula instead of calculating each term. Since b_n is the sum of a geometric sequence and an arithmetic sequence, perhaps I can separate the sums.So, S_5 = sum_{n=1 to 5} b_n = sum_{n=1 to 5} (a_n + n) = sum_{n=1 to 5} a_n + sum_{n=1 to 5} n.We know sum_{n=1 to 5} a_n is the sum of the first 5 terms of a geometric sequence with a_1 = 1 and q = 2.The formula for the sum of the first k terms of a geometric sequence is S_k = a_1*(q^k - 1)/(q - 1).So, S_5 for {a_n} is (2^5 - 1)/(2 - 1) = (32 - 1)/1 = 31.Then, sum_{n=1 to 5} n is the sum of the first 5 natural numbers, which is 5*(5 + 1)/2 = 15.So, S_5 = 31 + 15 = 46.Yep, same result. So, that's correct.Wait, let me double-check my calculations just to be sure.Calculating each b_n:- b_1: 2^{0} + 1 = 1 + 1 = 2- b_2: 2^{1} + 2 = 2 + 2 = 4- b_3: 2^{2} + 3 = 4 + 3 = 7- b_4: 2^{3} + 4 = 8 + 4 = 12- b_5: 2^{4} + 5 = 16 + 5 = 21Adding them up: 2 + 4 = 6; 6 + 7 = 13; 13 + 12 = 25; 25 + 21 = 46. Correct.Alternatively, using the formula: sum of a_n is 31, sum of n is 15, total 46. Correct.So, I think that's solid.**Final Answer**(1) The first term is boxed{1} and the nth term is boxed{2^{n-1}}.(2) The sum of the first 5 terms of the sequence {b_n} is boxed{46}.

answer：Okay, so I have this problem here: Compute (2 cos 20° + 2i sin 20°)^6. Hmm, I need to figure out how to solve this. Let me think. I remember that when dealing with complex numbers, especially in polar form, there's something called De Moivre's Theorem. Maybe that can help me here.First, let me recall what De Moivre's Theorem says. It states that for any complex number in the form r(cos θ + i sin θ), raising it to the power of n gives r^n (cos(nθ) + i sin(nθ)). So, if I can express the given complex number in that polar form, I can apply the theorem directly.Looking at the expression, it's 2 cos 20° + 2i sin 20°. That looks similar to r(cos θ + i sin θ), where r is the modulus and θ is the argument. In this case, r is 2, right? Because both the cosine and sine terms are multiplied by 2. And θ is 20°, since that's the angle given.So, if I rewrite the expression, it's 2(cos 20° + i sin 20°). Perfect, that's the standard polar form. Now, I need to raise this to the 6th power. According to De Moivre's Theorem, I can do this by raising the modulus to the 6th power and multiplying the angle by 6.Let me write that out step by step. The modulus r is 2, so r^6 would be 2^6. Let me compute that: 2^6 is 64. Okay, so the modulus part is 64.Next, the angle θ is 20°, so 6θ would be 6 * 20°, which is 120°. So, the angle after raising to the 6th power is 120°. Therefore, according to De Moivre's Theorem, the expression becomes 64(cos 120° + i sin 120°).Now, I need to convert this back into rectangular form, which is a + bi. To do that, I need to compute the cosine and sine of 120° and then multiply them by 64.Let me recall the values of cosine and sine for 120°. I know that 120° is in the second quadrant, where cosine is negative and sine is positive. Also, 120° is equal to 180° - 60°, so it's related to the 60° reference angle.The cosine of 60° is 0.5, so the cosine of 120° should be -0.5. Similarly, the sine of 60° is (√3)/2, so the sine of 120° is also (√3)/2.Let me write that down:cos 120° = -1/2sin 120° = √3/2So, plugging these into the expression:64(cos 120° + i sin 120°) = 64(-1/2 + i(√3)/2)Now, let's compute each part separately. First, 64 multiplied by -1/2. That's 64 * (-1/2) = -32. Next, 64 multiplied by (√3)/2. That's 64 * (√3)/2 = 32√3.Putting it all together, the rectangular form is -32 + 32√3 i.Wait, let me double-check my calculations to make sure I didn't make a mistake. So, 2^6 is 64, that's correct. 6 * 20° is 120°, that's right. Cos 120° is indeed -1/2, and sin 120° is √3/2. Multiplying 64 by -1/2 gives -32, and 64 times √3/2 is 32√3. So, yes, that seems correct.Alternatively, just to be thorough, I can compute this using another method, maybe by expanding the binomial, but that would be more complicated. Since De Moivre's Theorem is straightforward here, I think my answer is correct.So, the final result in rectangular form is -32 + 32√3 i.**Final Answer**The result is boxed{-32 + 32sqrt{3}i}.