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Home » Let the integers be x and x+2. Then, x(x+2) = 143. - Midis

Let the integers be x and x+2. Then, x(x+2) = 143. - Midis

📅 March 11, 2026 👤 scraface

Mar 11, 2026

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📌 \lim_{n \to \infty} a_n = \lim_{n \to \infty} \sum_{k=1}^{n} \left( \frac{k}{n} - \frac{1}{3} \left( \frac{k}{n} \right)^3 \right)

📌 This is a Riemann sum. Split the sum:

📌 a_n = \sum_{k=1}^{n} \frac{k}{n} - \frac{1}{3} \sum_{k=1}^{n} \left( \frac{k}{n} \right)^3

📌 = \frac{1}{n} \sum_{k=1}^{n} k - \frac{1}{3n^3} \sum_{k=1}^{n} k^3

📌 Use formulas:

📌 $ \sum_{k=1}^{n} k = \frac{n(n+1)}{2} $

📌 $ \sum_{k=1}^{n} k^3 = \left( \frac{n(n+1)}{2} \right)^2 $

📌 a_n = \frac{1}{n} \cdot \frac{n(n+1)}{2} - \frac{1}{3n^3} \cdot \left( \frac{n(n+1)}{2} \right)^2

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So for each pair, 30 valid sequences.
Total favorable: \(15 \times 30 = 450\)
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