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\textTime \approx 141 \text minutes - RoadRUNNER Motorcycle Touring & Travel Magazine

📅 April 30, 2026 👤 scraface

Apr 30, 2026

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📌 Thus, the length of the hypotenuse is \(\boxed{10}\).**Question:

📌 To compute the sum \(\sum_{k=1}^{50} \frac{1}{k(k+1)}\), we can use partial fraction decomposition. We start by expressing \(\frac{1}{k(k+1)}\) in a simpler form:

📌 Multiplying through by \(k(k+1)\) gives:

📌 = A(k+1) + Bk = Ak + A + Bk

📌 = (A + B)k + A

📌 For the equation to hold for all \(k\), we set up the system:

📌 A + B = 0 \quad \text{and} \quad A = 1

📌 Solving this, we find \(A = 1\) and \(B = -1\). Thus:

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