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Time \( t \) = 3 years - Midis

📅 March 11, 2026 👤 scraface

Mar 11, 2026

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📌 \frac{1}{k(k+2)} = \frac{1/2}{k} - \frac{1/2}{k+2} = \frac{1}{2} \left( \frac{1}{k} - \frac{1}{k+2} \right)

📌 \sum_{k=1}^{50} \frac{1}{k(k+2)} = \frac{1}{2} \sum_{k=1}^{50} \left( \frac{1}{k} - \frac{1}{k+2} \right)

📌 \frac{1}{2} \left[ \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) \right]

📌 Most terms cancel. The surviving positive terms are \(\frac{1}{1}, \frac{1}{2}\), and the surviving negative terms are \(-\frac{1}{51}, -\frac{1}{52}\) (since \(k+2 = 51, 52\) have no positive counterparts to cancel them).

📌 \frac{1}{2} \left( 1 + \frac{1}{2} - \frac{1}{51} - \frac{1}{52} \right) = \frac{1}{2} \left( \frac{3}{2} - \left( \frac{1}{51} + \frac{1}{52} \right) \right)

📌 Compute \(\frac{1}{51} + \frac{1}{52} = \frac{52 + 51}{51 \cdot 52} = \frac{103}{2652}\)

📌 \frac{1}{2} \left( \frac{3}{2} - \frac{103}{2652} \right) = \frac{1}{2} \left( \frac{3978 - 103}{2652} \right) = \frac{1}{2} \cdot \frac{3875}{2652} = \frac{3875}{5304}

📌 Thus, the exact value of the sum is \(\boxed{\dfrac{3875}{5304}}\).

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