💡Alright, so I have this problem about a sequence of positive terms. The first term is 1, and there's this recurrence relation: (n+1)a_{n+1}^2 - n a_n^2 + a_{n+1}a_n = 0 for n in the natural numbers starting from 1. I need to find the general formula for this sequence using induction. Hmm, okay, let me try to figure this out step by step.First, let's write down what we know. The sequence starts with a_1 = 1. Then, for each n ≥ 1, the relation (n+1)a_{n+1}^2 - n a_n^2 + a_{n+1}a_n = 0 holds. I need to find a general formula for a_n.Looking at the recurrence relation, it's a quadratic equation in terms of a_{n+1}. Maybe I can solve for a_{n+1} in terms of a_n. Let me rearrange the equation:(n+1)a_{n+1}^2 + a_{n+1}a_n - n a_n^2 = 0.This is a quadratic equation of the form A x^2 + B x + C = 0, where x is a_{n+1}, A = (n+1), B = a_n, and C = -n a_n^2.Using the quadratic formula, x = [-B ± sqrt(B^2 - 4AC)] / (2A). Plugging in the values:a_{n+1} = [-a_n ± sqrt(a_n^2 - 4*(n+1)*(-n a_n^2))]/(2*(n+1)).Simplify the discriminant:sqrt(a_n^2 - 4*(n+1)*(-n a_n^2)) = sqrt(a_n^2 + 4n(n+1)a_n^2) = sqrt(a_n^2*(1 + 4n(n+1))).Wait, that seems a bit complicated. Maybe there's a simpler way to factor the original equation. Let me try factoring:(n+1)a_{n+1}^2 + a_{n+1}a_n - n a_n^2 = 0.Looking at this, maybe I can factor it as (something with a_{n+1} and a_n). Let me try:Let's see, if I factor out a_{n+1} from the first two terms:a_{n+1}[(n+1)a_{n+1} + a_n] - n a_n^2 = 0.Hmm, not sure if that helps. Alternatively, maybe factor by grouping. Let me try:Group the first and third terms: (n+1)a_{n+1}^2 - n a_n^2, and then the middle term: + a_{n+1}a_n.So, (n+1)a_{n+1}^2 - n a_n^2 + a_{n+1}a_n = 0.Wait, maybe I can write this as:(n+1)a_{n+1}^2 + a_{n+1}a_n = n a_n^2.Then, factor out a_{n+1} from the left side:a_{n+1}[(n+1)a_{n+1} + a_n] = n a_n^2.Hmm, still not obvious. Maybe I can divide both sides by a_n^2 to make it in terms of ratios. Let me try that:Divide both sides by a_n^2:(n+1)(a_{n+1}/a_n)^2 + (a_{n+1}/a_n) - n = 0.Let me set r_n = a_{n+1}/a_n. Then the equation becomes:(n+1)r_n^2 + r_n - n = 0.Now, this is a quadratic in r_n:(n+1)r_n^2 + r_n - n = 0.Let me solve for r_n using the quadratic formula:r_n = [-1 ± sqrt(1 + 4(n+1)*n)] / [2(n+1)].Simplify the discriminant:sqrt(1 + 4n(n+1)) = sqrt(1 + 4n^2 + 4n) = sqrt(4n^2 + 4n + 1) = sqrt((2n + 1)^2) = 2n + 1.So, r_n = [-1 ± (2n + 1)] / [2(n+1)].Now, since the sequence consists of positive terms, r_n must be positive. So, we discard the negative solution:r_n = [-1 + (2n + 1)] / [2(n+1)] = (2n) / [2(n+1)] = n / (n+1).Therefore, a_{n+1}/a_n = n / (n+1).So, a_{n+1} = (n / (n+1)) a_n.This is a recursive relation where each term is a fraction of the previous term. To find the general formula, we can express a_n in terms of a_1.Starting from a_1 = 1:a_2 = (1/2) a_1 = 1/2.a_3 = (2/3) a_2 = (2/3)(1/2) = 1/3.a_4 = (3/4) a_3 = (3/4)(1/3) = 1/4.Hmm, I see a pattern here. It seems like a_n = 1/n.Let me test this conjecture with the recurrence relation.Assume a_n = 1/n. Then a_{n+1} = 1/(n+1).Plug into the original recurrence:(n+1)(1/(n+1))^2 - n(1/n)^2 + (1/(n+1))(1/n) = ?Simplify each term:(n+1)(1/(n+1)^2) = (n+1)/(n+1)^2 = 1/(n+1).-n(1/n^2) = -n/n^2 = -1/n.(1/(n+1))(1/n) = 1/(n(n+1)).So, total is 1/(n+1) - 1/n + 1/(n(n+1)).Combine the terms:1/(n+1) + 1/(n(n+1)) - 1/n = [n + 1 - n(n+1)] / [n(n+1)].Wait, let me compute it step by step:First, 1/(n+1) - 1/n = (n - (n+1)) / [n(n+1)] = (-1) / [n(n+1)].Then, adding 1/(n(n+1)) gives:(-1)/[n(n+1)] + 1/[n(n+1)] = 0.So, the left-hand side equals zero, which matches the right-hand side. Therefore, the conjecture a_n = 1/n satisfies the recurrence relation.To be thorough, let's use mathematical induction to prove that a_n = 1/n for all n ≥ 1.**Base Case:** For n = 1, a_1 = 1, which matches 1/1 = 1. So, the base case holds.**Inductive Step:** Assume that for some k ≥ 1, a_k = 1/k. We need to show that a_{k+1} = 1/(k+1).From the recurrence relation, we have:a_{k+1} = (k / (k+1)) a_k.Using the inductive hypothesis, a_k = 1/k, so:a_{k+1} = (k / (k+1)) * (1/k) = 1/(k+1).Thus, if a_k = 1/k, then a_{k+1} = 1/(k+1). Therefore, by induction, a_n = 1/n for all n ≥ 1.So, the general formula for the sequence is a_n = 1/n.

💡First, I need to simplify the left side of the equation by multiplying the fractions (frac{4}{7}) and (frac{1}{5}).Next, I'll isolate (x) by multiplying both sides of the equation by the reciprocal of the product of the two fractions.Finally, I'll perform the multiplication on the right side to find the value of (x).

💡Okay, so I have these two math problems to solve, and I need to figure them out step by step. Let me start with the first one.**Problem 1:** Calculate √(1/6) × √96 ÷ √6.Hmm, okay. So, I need to multiply the square root of 1/6 by the square root of 96 and then divide that by the square root of 6. I remember that when you multiply square roots, you can combine them under a single square root. So, maybe I can write this as √(1/6 × 96) ÷ √6.Let me compute what's inside the first square root first: 1/6 times 96. Well, 96 divided by 6 is 16, right? So, that simplifies to √16 ÷ √6. √16 is 4, so now I have 4 ÷ √6. But I think it's better to rationalize the denominator. To do that, I can multiply both the numerator and the denominator by √6. So, 4 × √6 over √6 × √6, which is 4√6 over 6. Simplifying that, I can divide numerator and denominator by 2, giving me 2√6 over 3. So, the first problem simplifies to 2√6/3.Wait, did I do that right? Let me check. Starting with √(1/6) × √96 ÷ √6. Combining the square roots: √[(1/6) × 96] ÷ √6. Inside the square root, 1/6 × 96 is indeed 16, so √16 is 4. Then, 4 ÷ √6. Rationalizing gives 4√6/6, which simplifies to 2√6/3. Yeah, that seems correct.**Problem 2:** Calculate √80 - √8 - √45 + 4√(1/2).Alright, this one looks a bit more involved. I need to simplify each square root term individually and then combine them. Let's start with √80. I know that 80 can be broken down into 16 × 5, and since √16 is 4, √80 is 4√5.Next, √8. That can be written as √(4×2), and since √4 is 2, that becomes 2√2.Then, √45. Breaking that down, 45 is 9×5, so √45 is 3√5.Lastly, 4√(1/2). I think I can simplify √(1/2) first. √(1/2) is the same as √1 over √2, which is 1/√2. But it's better to rationalize the denominator, so multiplying numerator and denominator by √2 gives √2/2. Therefore, 4 times √2/2 is (4×√2)/2, which simplifies to 2√2.Now, substituting all these back into the original expression:√80 - √8 - √45 + 4√(1/2) becomes 4√5 - 2√2 - 3√5 + 2√2.Let me combine like terms. The terms with √5 are 4√5 and -3√5. Combining those gives (4 - 3)√5, which is √5.The terms with √2 are -2√2 and +2√2. Combining those gives (-2 + 2)√2, which is 0.So, the entire expression simplifies to √5 + 0, which is just √5.Wait, let me double-check that. Starting with 4√5 - 2√2 - 3√5 + 2√2. Combining the √5 terms: 4√5 - 3√5 is √5. Combining the √2 terms: -2√2 + 2√2 is 0. Yep, that leaves us with √5.So, both problems seem to have simplified nicely. The first one is 2√6 over 3, and the second one is just √5. I think I've got it!

💡Okay, so I have this function f(x) = (2πx)^2, and I need to find its derivative, f'(x). Hmm, derivatives can sometimes be tricky, especially when constants are involved, but let's take it step by step.First, I remember that when you have a function raised to a power, like (something)^2, you can use the chain rule to find the derivative. The chain rule says that if you have a composite function, you take the derivative of the outer function and multiply it by the derivative of the inner function. So, in this case, the outer function is something squared, and the inner function is 2πx.Let me write that down to make it clearer. Let’s say u = 2πx, so the function becomes f(x) = u^2. Now, to find f'(x), I need to find the derivative of u^2 with respect to u and then multiply it by the derivative of u with respect to x.The derivative of u^2 with respect to u is straightforward. Using the power rule, which says that the derivative of u^n is n*u^(n-1), here n is 2, so the derivative is 2u.Now, I need to find the derivative of the inner function u = 2πx with respect to x. Since 2π is a constant, the derivative of 2πx with respect to x is just 2π.So, putting it all together, the derivative f'(x) is the derivative of the outer function times the derivative of the inner function. That would be 2u * 2π.But wait, u is 2πx, so I need to substitute that back in. So, 2*(2πx) * 2π. Let me calculate that step by step.First, 2*(2πx) is 4πx. Then, multiplying that by 2π gives 4πx * 2π. Multiplying the constants first, 4*2 is 8, and π*π is π squared. So, altogether, that's 8π²x.Let me double-check my steps to make sure I didn't make a mistake. I used the chain rule correctly: derivative of the outer function is 2u, derivative of the inner function is 2π, multiplied them together to get 4πu, then substituted u back in to get 8π²x. That seems right.Alternatively, I could have expanded the function first before taking the derivative. Let's try that approach to verify.Expanding (2πx)^2 gives 4π²x². Now, taking the derivative of 4π²x² with respect to x, using the power rule again, brings down the exponent as a coefficient, so 2*4π²x, which simplifies to 8π²x. That's the same result as before, so that confirms that my answer is correct.I think I'm confident now that the derivative of f(x) = (2πx)^2 is f'(x) = 8π²x. Looking back at the options provided:A) 4πxB) 4π²xC) 8π²xD) 16πxOption C matches my result, so that must be the correct answer.

💡Alright, so I've got this function f(x) defined piecewise. It's (x-1)^2 when x is between 0 and 2, and 2(x-2) when x is 2 or greater. The problem says that f(a) equals f(a+2), and I need to find f(a + sqrt(3)). The options are A: 0, B: 4sqrt(3), C: 0 or 4sqrt(3), and D: 4 - 2sqrt(3). Okay, let's break this down. First, I need to figure out what a is such that f(a) = f(a+2). Since the function is defined differently in two intervals, I should consider where a and a+2 fall in those intervals.Let me think about the possible ranges for a. If a is less than 2, then a+2 would be greater than or equal to 2. That makes sense because if a is, say, 1, then a+2 is 3, which is definitely in the second part of the function. So, if a is between 0 and 2, then a+2 is between 2 and 4, which is in the second piece of the function.So, f(a) would be (a - 1)^2 because a is between 0 and 2. And f(a+2) would be 2*(a+2 - 2) = 2a because a+2 is greater than or equal to 2. So, setting these equal: (a - 1)^2 = 2a.Let me write that equation down: (a - 1)^2 = 2a. Expanding the left side, I get a^2 - 2a + 1 = 2a. Bringing all terms to one side: a^2 - 4a + 1 = 0.Now, I need to solve this quadratic equation. Using the quadratic formula, a = [4 ± sqrt(16 - 4*1*1)] / 2 = [4 ± sqrt(12)] / 2 = [4 ± 2*sqrt(3)] / 2 = 2 ± sqrt(3).So, a could be 2 + sqrt(3) or 2 - sqrt(3). But wait, earlier I considered that a is between 0 and 2 because a+2 is greater than or equal to 2. So, a must be less than 2. Therefore, 2 + sqrt(3) is approximately 2 + 1.732 = 3.732, which is greater than 2, so that's not valid. The only valid solution is a = 2 - sqrt(3).Alright, so a is 2 - sqrt(3). Now, I need to find f(a + sqrt(3)). Let's compute a + sqrt(3): (2 - sqrt(3)) + sqrt(3) = 2. So, f(a + sqrt(3)) is f(2).Looking back at the function definition, when x is greater than or equal to 2, f(x) = 2(x - 2). So, f(2) = 2*(2 - 2) = 2*0 = 0. Therefore, f(a + sqrt(3)) is 0.Wait, let me double-check. If a = 2 - sqrt(3), then a + sqrt(3) is indeed 2. And f(2) is 0. So, that seems correct.But just to make sure, let me go through the steps again. Starting with f(a) = f(a+2). Since a is between 0 and 2, f(a) is (a - 1)^2, and f(a+2) is 2a. Setting them equal: (a - 1)^2 = 2a. Expanding: a^2 - 2a + 1 = 2a. Moving terms: a^2 - 4a + 1 = 0. Solving: a = [4 ± sqrt(16 - 4)] / 2 = [4 ± sqrt(12)] / 2 = 2 ± sqrt(3). Since a must be less than 2, a = 2 - sqrt(3). Then a + sqrt(3) = 2, so f(2) = 0. Yep, that all checks out.I think I'm confident that the answer is 0, which is option A.

💡First, I need to determine the type of sequence this pattern follows. Since each stage adds a fixed number of sticks, specifically four, this indicates an arithmetic sequence.Next, I'll identify the first term of the sequence. From the problem, the first stage has 4 sticks, so the first term ( a_1 ) is 4.The common difference ( d ) is given as 4, as four sticks are added at each successive stage.To find the number of sticks in the 150th stage, I'll use the formula for the nth term of an arithmetic sequence:[a_n = a_1 + (n-1)d]Substituting the known values:[a_{150} = 4 + (150-1) times 4][a_{150} = 4 + 149 times 4][a_{150} = 4 + 596][a_{150} = 600]Therefore, the arrangement for the 150th stage requires 600 sticks.