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Home » f(n) = \fracn^2 + 2n + 3n + 1. - Crosslake

f(n) = \fracn^2 + 2n + 3n + 1. - Crosslake

📅 March 1, 2026 👤 scraface

Mar 01, 2026

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📌 Compute \(\sum_{k=1}^{50} \frac{1}{k(k+1)}\).

📌 The given sum can be simplified using partial fraction decomposition:

📌 \frac{1}{k(k+1)} = \frac{1}{k} - \frac{1}{k+1}

📌 Thus, the sum becomes a telescoping series:

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

📌 Writing out the terms:

📌 \left( \frac{1}{1} - \frac{1}{2} \right) + \left( \frac{1}{2} - \frac{1}{3} \right) + \cdots + \left( \frac{1}{50} - \frac{1}{51} \right)

📌 Most terms cancel, leaving:

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