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📅 May 17, 2026 👤 scraface

May 17, 2026

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📌 \sum_{n=1}^{50} \frac{1}{n(n+2)} = \frac{1}{2} \sum_{n=1}^{50} \left( \frac{1}{n} - \frac{1}{n+2} \right).

📌 This is a telescoping sum. Write out the first few terms:

📌 \left( \frac{1}{1} - \frac{1}{3} \right) + \left( \frac{1}{2} - \frac{1}{4} \right) + \left( \frac{1}{3} - \frac{1}{5} \right) + \cdots + \left( \frac{1}{50} - \frac{1}{52} \right).

📌 \frac{1}{2} \left( 1 + \frac{1}{2} - \frac{1}{51} - \frac{1}{52} \right).

📌 Compute the sum:

📌 \frac{1}{2} \left( \frac{3}{2} - \frac{1}{51} - \frac{1}{52} \right).

📌 Find a common denominator for the subtracted terms:

📌 \frac{1}{51} + \frac{1}{52} = \frac{52 + 51}{51 \cdot 52} = \frac{103}{2652}.

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