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Home » From Baby to Elder: The Hidden Average Hamster Life Term You Can’t Ignore! - Leaselab

From Baby to Elder: The Hidden Average Hamster Life Term You Can’t Ignore! - Leaselab

📅 February 27, 2026 👤 scraface

Feb 27, 2026

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📌 Question: Compute $ \sum_{n=1}^{50} \frac{1}{n(n+2)} $.

📌 Solution: We begin by simplifying the general term using partial fractions:

📌 \frac{1}{n(n+2)} = \frac{A}{n} + \frac{B}{n+2}.

📌 Multiplying both sides by $ n(n+2) $:

📌 = A(n+2) + Bn.

📌 = An + 2A + Bn = (A + B)n + 2A.

📌 A + B = 0, \quad 2A = 1 \Rightarrow A = \frac{1}{2}, \quad B = -\frac{1}{2}.

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

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