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water softener system repair - inBeat

📅 May 3, 2026 👤 scraface

May 03, 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{...
\frac{1}{n(n+2)} = \frac{1}{2} \left( \frac{1}{n} - \frac{1}{n+2} \rig...
\sum_{n=1}^{50} \frac{1}{n(n+2)} = \frac{1}{2} \sum_{n=1}^{50} \left(...
This is a telescoping sum. Write out the first few terms:
\left( \frac{1}{1} - \frac{1}{3} \right) + \left( \frac{1}{2} - \frac{...
\frac{1}{2} \left( 1 + \frac{1}{2} - \frac{1}{51} - \frac{1}{52} \righ...
Compute the sum:
\frac{1}{2} \left( \frac{3}{2} - \frac{1}{51} - \frac{1}{52} \right).

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