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Home » No More Mediocrity: The Medium Length Haircut That’ll Turn Eyes Everywhere! - Dyverse

No More Mediocrity: The Medium Length Haircut That’ll Turn Eyes Everywhere! - Dyverse

📅 March 11, 2026 👤 scraface

Mar 11, 2026

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📌 = 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).

📌 \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).

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