💡Okay, so I've got this problem here where I need to find the value of (frac{a + b}{a - b}) given that (frac{frac{1}{a} + frac{1}{b}}{frac{1}{a} - frac{1}{b}} = 1001). Hmm, let me try to figure this out step by step.First, let me write down what's given:[frac{frac{1}{a} + frac{1}{b}}{frac{1}{a} - frac{1}{b}} = 1001]I think the first thing I should do is simplify this complex fraction. Maybe if I combine the terms in the numerator and the denominator, it will make things clearer.Starting with the numerator of the big fraction: (frac{1}{a} + frac{1}{b}). To add these two fractions, I need a common denominator, which would be (ab). So, adding them together:[frac{1}{a} + frac{1}{b} = frac{b}{ab} + frac{a}{ab} = frac{a + b}{ab}]Okay, that simplifies the numerator. Now, let's do the same for the denominator of the big fraction: (frac{1}{a} - frac{1}{b}). Again, common denominator is (ab):[frac{1}{a} - frac{1}{b} = frac{b}{ab} - frac{a}{ab} = frac{b - a}{ab}]So now, substituting these back into the original equation, we have:[frac{frac{a + b}{ab}}{frac{b - a}{ab}} = 1001]Hmm, since both the numerator and the denominator have the same denominator (ab), they should cancel out when we divide them. Let me write that out:[frac{a + b}{ab} div frac{b - a}{ab} = frac{a + b}{ab} times frac{ab}{b - a} = frac{a + b}{b - a}]So, simplifying that, the (ab) terms cancel out, leaving us with:[frac{a + b}{b - a} = 1001]Wait a second, the problem is asking for (frac{a + b}{a - b}), not (frac{a + b}{b - a}). I notice that (frac{a + b}{b - a}) is actually the negative of (frac{a + b}{a - b}) because (b - a = -(a - b)). Let me write that:[frac{a + b}{b - a} = -frac{a + b}{a - b}]So, substituting back into our equation:[-frac{a + b}{a - b} = 1001]To solve for (frac{a + b}{a - b}), I can multiply both sides of the equation by -1:[frac{a + b}{a - b} = -1001]Wait, but looking at the answer choices, option A is -1001, which would make sense here. But hold on, let me double-check my steps to make sure I didn't make a mistake.Starting from the beginning:1. I combined (frac{1}{a} + frac{1}{b}) correctly to get (frac{a + b}{ab}).2. Similarly, (frac{1}{a} - frac{1}{b}) became (frac{b - a}{ab}).3. Dividing these two gives (frac{a + b}{b - a}), which is correct.4. Recognizing that (frac{a + b}{b - a}) is equal to (-frac{a + b}{a - b}) is also correct.5. Therefore, setting (-frac{a + b}{a - b} = 1001) leads to (frac{a + b}{a - b} = -1001).So, according to this, the answer should be -1001, which is option A. But wait, let me check the original problem again to make sure I didn't misread anything.The problem states:[frac{frac{1}{a} + frac{1}{b}}{frac{1}{a} - frac{1}{b}} = 1001]Yes, that's correct. And we're supposed to find (frac{a + b}{a - b}). So, according to my calculations, it's -1001. But let me think again—maybe I made a sign error somewhere.Looking back at the step where I simplified the denominator:[frac{1}{a} - frac{1}{b} = frac{b - a}{ab}]Yes, that's correct. So when I divide the two fractions, I get:[frac{frac{a + b}{ab}}{frac{b - a}{ab}} = frac{a + b}{b - a}]Which is equal to (-frac{a + b}{a - b}). So, setting that equal to 1001 gives:[-frac{a + b}{a - b} = 1001 implies frac{a + b}{a - b} = -1001]So, that seems consistent. Therefore, the answer should be -1001, which is option A.But wait, let me consider another approach to verify. Maybe cross-multiplying from the beginning.Given:[frac{frac{1}{a} + frac{1}{b}}{frac{1}{a} - frac{1}{b}} = 1001]Cross-multiplying:[frac{1}{a} + frac{1}{b} = 1001 left( frac{1}{a} - frac{1}{b} right)]Expanding the right side:[frac{1}{a} + frac{1}{b} = frac{1001}{a} - frac{1001}{b}]Now, let's bring all terms to one side:[frac{1}{a} + frac{1}{b} - frac{1001}{a} + frac{1001}{b} = 0]Combine like terms:[left( frac{1 - 1001}{a} right) + left( frac{1 + 1001}{b} right) = 0]Simplify the coefficients:[frac{-1000}{a} + frac{1002}{b} = 0]Move one term to the other side:[frac{1002}{b} = frac{1000}{a}]Cross-multiplying:[1002a = 1000b]Divide both sides by 1000:[frac{1002}{1000}a = b]Simplify the fraction:[frac{501}{500}a = b]So, (b = frac{501}{500}a). Now, let's compute (frac{a + b}{a - b}):Substitute (b) with (frac{501}{500}a):[frac{a + frac{501}{500}a}{a - frac{501}{500}a}]Factor out (a) in numerator and denominator:[frac{a left(1 + frac{501}{500}right)}{a left(1 - frac{501}{500}right)} = frac{1 + frac{501}{500}}{1 - frac{501}{500}}]Simplify the fractions:[frac{frac{500 + 501}{500}}{frac{500 - 501}{500}} = frac{frac{1001}{500}}{frac{-1}{500}} = frac{1001}{-1} = -1001]So, again, I get (-1001). That confirms my earlier result. Therefore, the correct answer is A) -1001.Wait, but looking back at the initial problem, the options are:A) -1001B) -1/1001C) 1/1001D) 1E) 1001So, -1001 is option A, which matches my conclusion.Just to be thorough, let me consider if there's another way to approach this problem, maybe by letting (x = frac{a}{b}) or something like that.Let me set (x = frac{a}{b}). Then, (a = xb). Let's substitute this into the original equation:[frac{frac{1}{xb} + frac{1}{b}}{frac{1}{xb} - frac{1}{b}} = 1001]Simplify numerator and denominator:Numerator: (frac{1}{xb} + frac{1}{b} = frac{1 + x}{xb})Denominator: (frac{1}{xb} - frac{1}{b} = frac{1 - x}{xb})So, the equation becomes:[frac{frac{1 + x}{xb}}{frac{1 - x}{xb}} = frac{1 + x}{1 - x} = 1001]So, (frac{1 + x}{1 - x} = 1001). Solve for (x):Multiply both sides by (1 - x):[1 + x = 1001(1 - x)]Expand the right side:[1 + x = 1001 - 1001x]Bring all terms to one side:[x + 1001x = 1001 - 1][1002x = 1000][x = frac{1000}{1002} = frac{500}{501}]So, (x = frac{a}{b} = frac{500}{501}). Therefore, (a = frac{500}{501}b).Now, compute (frac{a + b}{a - b}):Substitute (a = frac{500}{501}b):[frac{frac{500}{501}b + b}{frac{500}{501}b - b} = frac{left(frac{500}{501} + 1right)b}{left(frac{500}{501} - 1right)b}]Simplify numerator and denominator:Numerator: (frac{500}{501} + 1 = frac{500 + 501}{501} = frac{1001}{501})Denominator: (frac{500}{501} - 1 = frac{500 - 501}{501} = frac{-1}{501})So, the fraction becomes:[frac{frac{1001}{501}}{frac{-1}{501}} = frac{1001}{-1} = -1001]Again, I arrive at the same answer, -1001. This further solidifies that option A is correct.Just to ensure there's no oversight, let me recap:1. Starting with the given equation, I simplified the complex fraction by finding common denominators.2. This led me to an expression involving (frac{a + b}{b - a}), which I recognized as the negative of (frac{a + b}{a - b}).3. Solving for (frac{a + b}{a - b}) gave me -1001.4. Cross-verifying by cross-multiplying and solving for (a) in terms of (b) (and vice versa) also led to the same result.5. A third method, introducing a ratio (x = frac{a}{b}), also confirmed the answer.Therefore, after multiple approaches and checks, I'm confident that the correct answer is indeed -1001, which is option A.**Final Answer**boxed{A}

💡First, I need to determine the total distance the train must travel to completely cross the platform. This is the sum of the length of the train and the length of the platform.Next, I'll convert the train's speed from kilometers per hour to meters per second to ensure the units are consistent for the calculation.Finally, I'll use the formula for time, which is distance divided by speed, to find out how long it takes for the train to cross the platform.

💡Okay, so I have this problem about constructing a 1/(2011²)-array and finding the sum of all its terms. Then, I need to express this sum as a fraction m/n where m and n are coprime, and find the remainder when m + n is divided by 2011. Hmm, that sounds a bit complicated, but let me try to break it down step by step.First, the problem defines a 1/p²-array. It says it's an infinite collection where each term in a row is 1/p² times the previous term, and the first entry of each row is 1/(2p²) times the first entry of the previous row. So, it's like a two-dimensional infinite array where each row is a geometric sequence with ratio 1/p², and each subsequent row starts with a term that's 1/(2p²) times the first term of the previous row.Alright, so for p = 2011, each row will have terms decreasing by a factor of 1/(2011²), and each new row starts with a term that's 1/(2*2011²) times the first term of the row above it. Got it.Now, I need to find the sum of all the terms in this array. Since it's an infinite array, I can model this as a double sum: sum over all rows and within each row, sum over all columns. So, the total sum S would be:S = sum_{r=1 to ∞} sum_{c=1 to ∞} a_{r,c}Where a_{r,c} is the term in the rth row and cth column.From the problem statement, each term a_{r,c} can be expressed as:a_{r,c} = (1/(2p²))^{r} * (1/p²)^{c-1}Wait, hold on. Let me think. The first term of each row is 1/(2p²) times the first term of the previous row. So, the first term of the first row is, let's say, a_{1,1} = 1/(2p²) * something? Wait, actually, if it's an infinite array, the first term of the first row must be some initial value. But the problem doesn't specify, so maybe I need to figure it out.Wait, actually, the problem says each succeeding term in a row is 1/p² times the previous term, so each row is a geometric series with ratio 1/p². Similarly, the first term of each row is 1/(2p²) times the first term of the previous row, so the first terms of each row form another geometric series with ratio 1/(2p²).Therefore, the first term of the first row is some value, say, a_{1,1}. Then, the first term of the second row is a_{2,1} = a_{1,1} * 1/(2p²). The first term of the third row is a_{3,1} = a_{2,1} * 1/(2p²) = a_{1,1} * (1/(2p²))², and so on.Similarly, within each row, the terms are a geometric sequence. So, in the first row, the terms are a_{1,1}, a_{1,2} = a_{1,1} * 1/p², a_{1,3} = a_{1,2} * 1/p² = a_{1,1} * (1/p²)², etc.Therefore, the general term a_{r,c} can be written as:a_{r,c} = a_{1,1} * (1/(2p²))^{r - 1} * (1/p²)^{c - 1}Wait, is that correct? Let me check. For the first row (r=1), the first term is a_{1,1}, so when r=1, it's a_{1,1} * (1/(2p²))^{0} * (1/p²)^{c - 1}, which is a_{1,1} * (1/p²)^{c - 1}. That seems right.But actually, the first term of each row is 1/(2p²) times the first term of the previous row. So, the first term of row r is a_{1,1} * (1/(2p²))^{r - 1}. Then, within each row, each term is 1/p² times the previous term, so the cth term in row r is a_{r,c} = a_{1,1} * (1/(2p²))^{r - 1} * (1/p²)^{c - 1}.Therefore, the total sum S is:S = sum_{r=1 to ∞} sum_{c=1 to ∞} a_{1,1} * (1/(2p²))^{r - 1} * (1/p²)^{c - 1}But wait, the problem didn't specify what a_{1,1} is. Hmm. Maybe I need to figure out a_{1,1} such that the array is properly defined. Alternatively, perhaps a_{1,1} is 1, but I need to confirm.Wait, actually, in the problem statement, it says "each succeeding term in a row is 1/p² times the previous term in the same row, and the first entry of each row is 1/(2p²) times the first entry of the previous row." So, it doesn't specify the first term of the first row. Maybe it's arbitrary? Or perhaps it's 1? Hmm.Wait, actually, in the initial problem statement, it says "Construct a 1/(2011²)-array." So, maybe the first term is 1? Or perhaps it's 1/(2p²) times the previous row's first term, but since there is no previous row before the first one, maybe the first term is 1? Or maybe it's 1/(2p²) times something? Hmm, this is a bit confusing.Wait, perhaps the first term of the first row is 1/(2p²) times the first term of the "previous" row, but since there is no row before the first row, maybe the first term is 1/(2p²) times 1? Or perhaps it's just 1? Hmm.Wait, maybe I can assume that the first term of the first row is 1. Let's try that. So, a_{1,1} = 1. Then, a_{2,1} = 1/(2p²), a_{3,1} = (1/(2p²))², and so on. Then, within each row, the terms are 1/p² times the previous term.Therefore, the general term would be a_{r,c} = (1/(2p²))^{r - 1} * (1/p²)^{c - 1}.So, the total sum S would be:S = sum_{r=1 to ∞} sum_{c=1 to ∞} (1/(2p²))^{r - 1} * (1/p²)^{c - 1}But actually, if a_{1,1} = 1, then the first term is 1, and the rest follow accordingly. Alternatively, if a_{1,1} is something else, say, k, then the sum would be k times the double sum above.But since the problem doesn't specify, maybe the first term is 1. Let me proceed with that assumption.So, S = sum_{r=1 to ∞} sum_{c=1 to ∞} (1/(2p²))^{r - 1} * (1/p²)^{c - 1}But actually, in the problem statement, it says "each succeeding term in a row is 1/p² times the previous term in the same row, and the first entry of each row is 1/(2p²) times the first entry of the previous row." So, if the first term of the first row is a_{1,1}, then the first term of the second row is a_{2,1} = a_{1,1} * 1/(2p²). Then, the first term of the third row is a_{3,1} = a_{2,1} * 1/(2p²) = a_{1,1} * (1/(2p²))², etc.Similarly, within each row, the terms are a_{r,1}, a_{r,2} = a_{r,1} * 1/p², a_{r,3} = a_{r,2} * 1/p² = a_{r,1} * (1/p²)², etc.Therefore, the general term is a_{r,c} = a_{1,1} * (1/(2p²))^{r - 1} * (1/p²)^{c - 1}So, the total sum S is:S = sum_{r=1 to ∞} sum_{c=1 to ∞} a_{1,1} * (1/(2p²))^{r - 1} * (1/p²)^{c - 1}But since a_{1,1} is the first term, and the problem doesn't specify it, perhaps it's 1? Or maybe it's something else. Wait, actually, if we think about it, the first term of the first row is a_{1,1}, and the first term of the second row is a_{2,1} = a_{1,1} * 1/(2p²). So, if we set a_{1,1} = 1, then the first term is 1, and the rest follow. Alternatively, if we set a_{1,1} = k, then the sum would be k times the double sum.But since the problem is asking for the sum of all its terms, and it doesn't specify the starting term, maybe the starting term is 1. Let me assume that a_{1,1} = 1.Therefore, S = sum_{r=1 to ∞} sum_{c=1 to ∞} (1/(2p²))^{r - 1} * (1/p²)^{c - 1}Wait, but actually, if a_{1,1} = 1, then the first term is 1, and the rest are scaled accordingly. So, the sum would be:S = sum_{r=1 to ∞} [sum_{c=1 to ∞} (1/p²)^{c - 1}] * (1/(2p²))^{r - 1}But wait, that's not quite right. Because for each row r, the sum over c is a geometric series with ratio 1/p², starting from c=1, so the sum is:sum_{c=1 to ∞} (1/p²)^{c - 1} = 1 / (1 - 1/p²) = p² / (p² - 1)Similarly, the sum over r is a geometric series with ratio 1/(2p²), starting from r=1, so the sum is:sum_{r=1 to ∞} (1/(2p²))^{r - 1} = 1 / (1 - 1/(2p²)) = 2p² / (2p² - 1)Therefore, the total sum S is:S = [2p² / (2p² - 1)] * [p² / (p² - 1)] = (2p² * p²) / [(2p² - 1)(p² - 1)] = 2p⁴ / [(2p² - 1)(p² - 1)]So, substituting p = 2011, we get:S = 2*(2011)^4 / [(2*(2011)^2 - 1)((2011)^2 - 1)]Now, the problem says to express this sum as m/n where m and n are coprime positive integers, and then find m + n modulo 2011.So, first, let's compute the numerator and denominator.Numerator: 2*(2011)^4Denominator: (2*(2011)^2 - 1)*(2011^2 - 1)Let me compute each part step by step.First, compute 2011^2:2011^2 = 2011*2011. Let me compute that.2011*2000 = 4,022,0002011*11 = 22,121So, 4,022,000 + 22,121 = 4,044,121So, 2011^2 = 4,044,121Therefore, 2*(2011)^2 = 2*4,044,121 = 8,088,242So, 2*(2011)^2 - 1 = 8,088,242 - 1 = 8,088,241Similarly, 2011^2 - 1 = 4,044,121 - 1 = 4,044,120Therefore, the denominator is 8,088,241 * 4,044,120So, the fraction is:2*(2011)^4 / (8,088,241 * 4,044,120)But 2011^4 is (2011^2)^2 = (4,044,121)^2Let me compute 4,044,121^2. Hmm, that's a big number. Maybe I don't need to compute it directly because I'm going to reduce the fraction m/n modulo 2011.Wait, actually, since we need to find m + n modulo 2011, perhaps we can simplify the fraction modulo 2011 without computing the actual large numbers.Let me think about that.First, note that 2011 is a prime number. That might be useful because we can use properties modulo primes.Given that, let's consider the fraction S = 2p⁴ / [(2p² - 1)(p² - 1)] where p = 2011.We can write this as:S = 2p⁴ / [(2p² - 1)(p² - 1)]We need to express this as m/n where m and n are coprime, then find m + n mod 2011.But instead of computing m and n directly, maybe we can compute m/n modulo 2011, but since m/n is a fraction, we need to compute it as m * n^{-1} mod 2011, but since we need m + n, perhaps we can find m and n modulo 2011.Wait, but m and n are integers, and we need m + n mod 2011. So, perhaps we can compute m mod 2011 and n mod 2011, then add them together.But to do that, we need to simplify the fraction S modulo 2011.Let me consider the numerator and denominator modulo 2011.First, note that p = 2011, so p ≡ 0 mod 2011.Therefore, p² ≡ 0 mod 2011, and p⁴ ≡ 0 mod 2011.Similarly, 2p² - 1 ≡ -1 mod 2011, because 2p² ≡ 0 mod 2011, so 2p² - 1 ≡ -1 mod 2011.Similarly, p² - 1 ≡ -1 mod 2011.Therefore, the denominator (2p² - 1)(p² - 1) ≡ (-1)*(-1) = 1 mod 2011.The numerator is 2p⁴ ≡ 0 mod 2011.So, S = 0 / 1 mod 2011, which is 0.Wait, but that can't be right because m and n are coprime, so n can't be 0 mod 2011. Hmm, maybe I need to think differently.Wait, actually, when reducing the fraction modulo 2011, we have to be careful because the denominator could be invertible modulo 2011.Let me try to compute S modulo 2011.Given S = 2p⁴ / [(2p² - 1)(p² - 1)]Since p = 2011, which is 0 mod 2011, let's substitute p = 0 in the expression.So, S ≡ 2*0⁴ / [(2*0² - 1)(0² - 1)] mod 2011Simplify numerator: 2*0 = 0Denominator: ( -1 )*( -1 ) = 1So, S ≡ 0 / 1 ≡ 0 mod 2011But S is a fraction m/n, and we're to compute m + n mod 2011. So, if S ≡ 0 mod 2011, does that mean m ≡ 0 mod 2011 and n ≡ 1 mod 2011? Or perhaps m ≡ 0 mod 2011 and n is invertible.Wait, but m and n are coprime, so if m ≡ 0 mod 2011, then n cannot be ≡ 0 mod 2011, because they are coprime. So, n must be invertible modulo 2011.Therefore, S = m/n ≡ 0 mod 2011 implies that m ≡ 0 mod 2011, and n ≡ k mod 2011 where k is invertible.Therefore, m + n ≡ 0 + k ≡ k mod 2011.But wait, earlier, I thought S ≡ 0 mod 2011, but actually, S is equal to 2p⁴ / [(2p² - 1)(p² - 1)]. Let me compute this fraction modulo 2011.But since p = 2011, which is 0 mod 2011, let's substitute p = 0 in the expression:Numerator: 2*(0)^4 = 0Denominator: (2*(0)^2 - 1)*(0^2 - 1) = (-1)*(-1) = 1So, S ≡ 0/1 ≡ 0 mod 2011.But S is equal to m/n, so m/n ≡ 0 mod 2011. Since n is invertible modulo 2011 (because m and n are coprime, and n cannot be 0 mod 2011), this implies that m ≡ 0 mod 2011.Therefore, m is a multiple of 2011, say m = 2011*k for some integer k.Then, m + n = 2011*k + n.Since m and n are coprime, n must be coprime to 2011, because if n had a factor of 2011, then m and n would share that factor, which contradicts them being coprime.Therefore, n is invertible modulo 2011, and m is 0 modulo 2011.Thus, m + n ≡ 0 + n ≡ n mod 2011.But we need to find m + n mod 2011. Since m = 2011*k, m + n = 2011*k + n ≡ n mod 2011.But we need to find n mod 2011.Wait, but how can we find n mod 2011? Because n is the denominator of the reduced fraction S = m/n.Alternatively, perhaps we can compute S modulo 2011 as 0, which implies that m ≡ 0 mod 2011, and n ≡ something mod 2011.But since m/n ≡ 0 mod 2011, and n is invertible, that means m ≡ 0 mod 2011, but n is not 0 mod 2011.Therefore, m + n ≡ 0 + n ≡ n mod 2011.But we need to find n mod 2011.Wait, perhaps we can compute n modulo 2011 by looking at the denominator.The denominator is (2p² - 1)(p² - 1). Let's compute this modulo 2011.As before, p = 2011 ≡ 0 mod 2011.So, 2p² ≡ 0 mod 2011, so 2p² - 1 ≡ -1 mod 2011.Similarly, p² - 1 ≡ -1 mod 2011.Therefore, the denominator ≡ (-1)*(-1) = 1 mod 2011.So, the denominator is congruent to 1 modulo 2011.Therefore, n ≡ 1 mod 2011.Therefore, m + n ≡ 0 + 1 ≡ 1 mod 2011.Wait, but earlier I thought m + n ≡ n mod 2011, which would be 1. But in the initial problem, the assistant's answer was 0. Hmm, maybe I made a mistake.Wait, let me double-check.We have S = 2p⁴ / [(2p² - 1)(p² - 1)].We need to express this as m/n where m and n are coprime.Then, m + n mod 2011.But when reducing S modulo 2011, we get S ≡ 0 mod 2011, which implies that m ≡ 0 mod 2011 and n ≡ 1 mod 2011.Therefore, m + n ≡ 0 + 1 ≡ 1 mod 2011.But the assistant's answer was 0. Hmm, maybe I'm missing something.Wait, perhaps I need to consider that m/n is equal to S, and m and n are coprime. So, m/n = S = 2p⁴ / [(2p² - 1)(p² - 1)].But since p = 2011, which is prime, let's factor the denominator:Denominator: (2p² - 1)(p² - 1) = (2p² - 1)(p - 1)(p + 1)So, the denominator factors into (2p² - 1)(p - 1)(p + 1).Now, let's see if 2p² - 1 shares any common factors with p.Since p is prime, and 2p² - 1 is 2p² - 1. Let's compute 2p² - 1 modulo p.2p² ≡ 0 mod p, so 2p² - 1 ≡ -1 mod p.Therefore, 2p² - 1 and p are coprime.Similarly, p - 1 and p are coprime, as are p + 1 and p.Therefore, the denominator is coprime to p, which is 2011.Therefore, in the fraction m/n = 2p⁴ / [(2p² - 1)(p² - 1)], since the denominator is coprime to p, and the numerator is 2p⁴, which is divisible by p, the fraction can be written as m/n where m is divisible by p, and n is coprime to p.Therefore, m = p * k, and n is coprime to p.Therefore, m + n ≡ 0 + n ≡ n mod p.But we need to find n mod p.But n is the denominator divided by the gcd of numerator and denominator.Wait, actually, m/n is in lowest terms, so n is equal to [(2p² - 1)(p² - 1)] divided by the gcd of numerator and denominator.But the numerator is 2p⁴, and the denominator is (2p² - 1)(p² - 1).We need to find gcd(2p⁴, (2p² - 1)(p² - 1)).Since p is prime, and 2p² - 1 and p² - 1 are both coprime to p, as we saw earlier, the gcd is 1.Therefore, m/n is already in lowest terms, so m = 2p⁴ and n = (2p² - 1)(p² - 1).Therefore, m + n = 2p⁴ + (2p² - 1)(p² - 1)We need to compute this modulo p, which is 2011.So, let's compute m + n mod p.First, compute 2p⁴ mod p.Since p divides p⁴, 2p⁴ ≡ 0 mod p.Next, compute (2p² - 1)(p² - 1) mod p.Again, p² ≡ 0 mod p, so:(2p² - 1) ≡ -1 mod p(p² - 1) ≡ -1 mod pTherefore, (2p² - 1)(p² - 1) ≡ (-1)*(-1) = 1 mod pTherefore, m + n ≡ 0 + 1 ≡ 1 mod pSo, m + n ≡ 1 mod 2011Therefore, the remainder when m + n is divided by 2011 is 1.Wait, but the assistant's answer was 0. Hmm, maybe I made a mistake in the reasoning.Wait, let me check the initial assumption. I assumed that a_{1,1} = 1, but perhaps that's not correct.Wait, the problem says "Construct a 1/(2011²)-array." It doesn't specify the starting term, so maybe the starting term is 1/(2p²) times the previous row's first term, but since there is no previous row, perhaps the first term is 1/(2p²) times 1? Or maybe it's 1/(2p²) times something else.Wait, actually, the problem defines the array as starting with the first row, so perhaps the first term is arbitrary. But in the initial problem statement, the assistant's solution assumed that the first term is 1/(2p²) times the previous row's first term, but since there is no previous row, perhaps the first term is 1/(2p²) times 1, making the first term 1/(2p²). Wait, but that would make the first term of the first row 1/(2p²), and then the first term of the second row would be 1/(2p²) * 1/(2p²) = 1/(2p²)^2, etc.Wait, but in the assistant's solution, they wrote the general term as (1/(2p²))^r * (1/p²)^c, which suggests that the first term of the first row is (1/(2p²))^1 * (1/p²)^1 = 1/(2p² * p²) = 1/(2p⁴). Hmm, that seems different from my initial assumption.Wait, perhaps I need to clarify the indexing. The assistant's solution wrote the general term as (1/(2p²))^r * (1/p²)^c, which would mean that for r=1 and c=1, the term is 1/(2p²) * 1/p² = 1/(2p⁴). So, the first term is 1/(2p⁴). Then, the first term of the first row is 1/(2p⁴), and the first term of the second row is 1/(2p²) times that, which is 1/(2p⁴) * 1/(2p²) = 1/(4p⁶), and so on.Wait, but that seems a bit odd because the first term of the first row is 1/(2p⁴), which is quite small. Alternatively, maybe the first term of the first row is 1, and then the first term of the second row is 1/(2p²), etc. So, the general term is (1/(2p²))^{r - 1} * (1/p²)^{c - 1}.Wait, perhaps the confusion is in the indexing. Let me try to clarify.If the first term of the first row is a_{1,1}, then the first term of the second row is a_{2,1} = a_{1,1} * 1/(2p²). Similarly, a_{3,1} = a_{2,1} * 1/(2p²) = a_{1,1} * (1/(2p²))², etc.Within each row, the terms are a_{r,1}, a_{r,2} = a_{r,1} * 1/p², a_{r,3} = a_{r,2} * 1/p² = a_{r,1} * (1/p²)², etc.Therefore, the general term is a_{r,c} = a_{1,1} * (1/(2p²))^{r - 1} * (1/p²)^{c - 1}So, the total sum S is:S = sum_{r=1 to ∞} sum_{c=1 to ∞} a_{1,1} * (1/(2p²))^{r - 1} * (1/p²)^{c - 1}If we set a_{1,1} = 1, then S becomes:S = sum_{r=1 to ∞} sum_{c=1 to ∞} (1/(2p²))^{r - 1} * (1/p²)^{c - 1}Which is equal to:sum_{r=1 to ∞} (1/(2p²))^{r - 1} * sum_{c=1 to ∞} (1/p²)^{c - 1}Each of these is a geometric series.The sum over c is sum_{c=1 to ∞} (1/p²)^{c - 1} = 1 / (1 - 1/p²) = p² / (p² - 1)Similarly, the sum over r is sum_{r=1 to ∞} (1/(2p²))^{r - 1} = 1 / (1 - 1/(2p²)) = 2p² / (2p² - 1)Therefore, S = (2p² / (2p² - 1)) * (p² / (p² - 1)) = 2p⁴ / [(2p² - 1)(p² - 1)]So, that's the same as before.Therefore, m/n = 2p⁴ / [(2p² - 1)(p² - 1)]Since p = 2011, m = 2*(2011)^4 and n = (2*(2011)^2 - 1)*(2011^2 - 1)We need to find m + n mod 2011.As before, since p = 2011, let's compute m + n modulo p.First, m = 2p⁴. Since p divides p⁴, m ≡ 0 mod p.Next, n = (2p² - 1)(p² - 1). Let's compute this modulo p.2p² ≡ 0 mod p, so 2p² - 1 ≡ -1 mod pSimilarly, p² - 1 ≡ -1 mod pTherefore, n ≡ (-1)*(-1) = 1 mod pTherefore, m + n ≡ 0 + 1 ≡ 1 mod pSo, the remainder when m + n is divided by 2011 is 1.Wait, but the assistant's answer was 0. Hmm, maybe I made a mistake in the initial assumption about a_{1,1}.Wait, in the assistant's solution, they wrote the general term as (1/(2p²))^r * (1/p²)^c, which would mean that a_{1,1} = (1/(2p²))^1 * (1/p²)^1 = 1/(2p⁴). So, the first term is 1/(2p⁴). Then, the first term of the second row is 1/(2p²) times that, which is 1/(2p⁴) * 1/(2p²) = 1/(4p⁶), etc.In that case, the sum S would be:S = sum_{r=1 to ∞} sum_{c=1 to ∞} (1/(2p²))^r * (1/p²)^cWhich is equal to:sum_{r=1 to ∞} (1/(2p²))^r * sum_{c=1 to ∞} (1/p²)^cEach of these is a geometric series starting from r=1 and c=1.So, sum_{r=1 to ∞} (1/(2p²))^r = (1/(2p²)) / (1 - 1/(2p²)) = 1 / (2p² - 1)Similarly, sum_{c=1 to ∞} (1/p²)^c = (1/p²) / (1 - 1/p²) = 1 / (p² - 1)Therefore, S = [1 / (2p² - 1)] * [1 / (p² - 1)] = 1 / [(2p² - 1)(p² - 1)]So, in this case, m/n = 1 / [(2p² - 1)(p² - 1)]Therefore, m = 1 and n = (2p² - 1)(p² - 1)So, m + n = 1 + (2p² - 1)(p² - 1)Now, compute m + n modulo p.Again, p = 2011.Compute (2p² - 1)(p² - 1) mod p.As before, 2p² ≡ 0 mod p, so 2p² - 1 ≡ -1 mod pSimilarly, p² - 1 ≡ -1 mod pTherefore, (2p² - 1)(p² - 1) ≡ (-1)*(-1) = 1 mod pTherefore, m + n = 1 + 1 = 2 mod pWait, that's different. So, in this case, m + n ≡ 2 mod 2011.But the assistant's answer was 0. Hmm, this is confusing.Wait, perhaps the initial term a_{1,1} is 1, so the sum is 2p⁴ / [(2p² - 1)(p² - 1)], and m + n ≡ 1 mod p.Alternatively, if a_{1,1} is 1/(2p²), then the sum is 1 / [(2p² - 1)(p² - 1)], and m + n ≡ 2 mod p.But the problem statement doesn't specify the starting term, so perhaps the assistant assumed a different starting term.Wait, let me read the problem statement again."A 1/p²-array is an infinite, structured collection of numbers where each succeeding term in a row is 1/p² times the previous term in the same row, and the first entry of each row is 1/(2p²) times the first entry of the previous row."It doesn't specify the first term of the first row, so perhaps it's arbitrary. But in the assistant's solution, they wrote the general term as (1/(2p²))^r * (1/p²)^c, which suggests that the first term is 1/(2p²) * 1/p² = 1/(2p⁴). So, the first term is 1/(2p⁴), and the sum is 1 / [(2p² - 1)(p² - 1)].Therefore, in that case, m = 1 and n = (2p² - 1)(p² - 1). Then, m + n = 1 + (2p² - 1)(p² - 1). As computed earlier, this is congruent to 2 mod p.But the assistant's answer was 0. Hmm, maybe I'm missing something.Wait, perhaps the problem is that the array is defined such that the first entry of each row is 1/(2p²) times the first entry of the previous row, but the first entry of the first row is 1. So, a_{1,1} = 1, a_{2,1} = 1/(2p²), a_{3,1} = 1/(2p²)^2, etc.Then, within each row, the terms are a_{r,1}, a_{r,2} = a_{r,1} * 1/p², etc.Therefore, the general term is a_{r,c} = (1/(2p²))^{r - 1} * (1/p²)^{c - 1}Then, the sum S is:S = sum_{r=1 to ∞} sum_{c=1 to ∞} (1/(2p²))^{r - 1} * (1/p²)^{c - 1}Which is equal to:sum_{r=1 to ∞} (1/(2p²))^{r - 1} * sum_{c=1 to ∞} (1/p²)^{c - 1}Each of these is a geometric series.Sum over c: sum_{c=1 to ∞} (1/p²)^{c - 1} = 1 / (1 - 1/p²) = p² / (p² - 1)Sum over r: sum_{r=1 to ∞} (1/(2p²))^{r - 1} = 1 / (1 - 1/(2p²)) = 2p² / (2p² - 1)Therefore, S = (2p² / (2p² - 1)) * (p² / (p² - 1)) = 2p⁴ / [(2p² - 1)(p² - 1)]So, m/n = 2p⁴ / [(2p² - 1)(p² - 1)]Then, m + n = 2p⁴ + (2p² - 1)(p² - 1)Compute this modulo p.2p⁴ ≡ 0 mod p(2p² - 1)(p² - 1) ≡ (-1)*(-1) = 1 mod pTherefore, m + n ≡ 0 + 1 ≡ 1 mod pSo, the remainder is 1.But the assistant's answer was 0. Hmm, maybe the assistant made a mistake.Wait, let me check the assistant's solution.The assistant wrote:"The value in the rth row and cth column of a 1/p²-array is given by (1/(2p²)^r)(1/(p²)^c).The overall sum of the terms in the array is:sum_{r=1}^∞ sum_{c=1}^∞ (1/(2p²)^r)(1/(p²)^c) = [sum_{r=1}^∞ 1/(2p²)^r][sum_{c=1}^∞ 1/(p²)^c] = [1/(1 - 1/(2p²))][1/(1 - 1/p²)] = 2p⁴ / [(2p² - 1)(p² - 1)]Substituting p = 2011, we find:2 * 2011⁴ / [(2 * 2011² - 1)(2011² - 1)]To find m/n, we rewrite m + n modulo 2011:m + n ≡ 2011⁴ + (2 * 2011² - 1)(2011² - 1) ≡ 0 mod 2011The final answer is 0."Wait, the assistant computed m + n as 2011⁴ + (2 * 2011² - 1)(2011² - 1). But in reality, m = 2 * 2011⁴ and n = (2 * 2011² - 1)(2011² - 1). So, m + n = 2 * 2011⁴ + (2 * 2011² - 1)(2011² - 1).But the assistant wrote m + n ≡ 2011⁴ + (2 * 2011² - 1)(2011² - 1) mod 2011, which is incorrect because m = 2 * 2011⁴, not 2011⁴.Therefore, the assistant made a mistake in the calculation of m + n modulo 2011.So, the correct calculation is:m + n = 2 * 2011⁴ + (2 * 2011² - 1)(2011² - 1)Compute this modulo 2011.First, 2 * 2011⁴ ≡ 0 mod 2011Next, compute (2 * 2011² - 1)(2011² - 1) mod 2011.As before, 2 * 2011² ≡ 0 mod 2011, so 2 * 2011² - 1 ≡ -1 mod 2011Similarly, 2011² - 1 ≡ -1 mod 2011Therefore, (2 * 2011² - 1)(2011² - 1) ≡ (-1)*(-1) = 1 mod 2011Therefore, m + n ≡ 0 + 1 ≡ 1 mod 2011So, the correct remainder is 1, not 0.Therefore, the assistant's answer was incorrect. The correct answer should be 1.But wait, in the initial problem statement, the assistant wrote:"To find m/n, we rewrite m + n modulo 2011:m + n ≡ 2011⁴ + (2 * 2011² - 1)(2011² - 1) ≡ 0 mod 2011"But as we saw, m = 2 * 2011⁴, not 2011⁴, so the correct expression is m + n = 2 * 2011⁴ + (2 * 2011² - 1)(2011² - 1), which modulo 2011 is 0 + 1 = 1.Therefore, the correct remainder is 1.So, the final answer should be 1, not 0.But the assistant's answer was 0, which is incorrect.Therefore, the correct answer is 1.

💡First, I need to determine the interest earned over the 5 years by subtracting the principal amount from the final amount.Next, I'll use the simple interest formula to find the rate of interest. The formula for simple interest is:[ text{SI} = frac{text{Principal} times text{Rate} times text{Time}}{100} ]I have the values for the simple interest, principal, and time. I'll plug these into the formula and solve for the rate.After calculating, I'll find that the rate of interest is 3.75%.

💡First, I need to find the least prime factor of (5^6 - 5^4). To simplify the expression, I'll factor out the common term (5^4).This gives me:[5^6 - 5^4 = 5^4 (5^2 - 1) = 5^4 times (25 - 1) = 5^4 times 24]Next, I'll factorize 24 to find its prime components:[24 = 2^3 times 3]Now, I have the prime factors of the expression:[5^4 times 2^3 times 3]Among these prime factors, the smallest one is 2. Therefore, the least prime factor of (5^6 - 5^4) is 2.

💡To determine the values of ( m ) and ( n ), we start by recognizing that for the terms ( 5x^{m+2}y^{3} ) and ( frac{1}{4}x^6y^{n+1} ) to be like terms, their corresponding exponents must be equal.1. **Equating the exponents of ( x ):** [ m + 2 = 6 ] Solving for ( m ): [ m = 6 - 2 = 4 ]2. **Equating the exponents of ( y ):** [ n + 1 = 3 ] Solving for ( n ): [ n = 3 - 1 = 2 ]With ( m = 4 ) and ( n = 2 ), we now calculate ( (-m)^3 + n^2 ):[(-4)^3 + 2^2 = -64 + 4 = -60]