= An + 2A + Bn = (A + B)n + 2A.
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- \frac{1}{n(n+2)} = \frac{A}{n} + \frac{B}{n+2}.
- Multiplying both sides by $ n(n+2) $:
- = A(n+2) + Bn.
- 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).
