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Home » Dare to Explore the Myriad Colors Phantom World—Colors You Never Thought Existed! - Crankk.io

Dare to Explore the Myriad Colors Phantom World—Colors You Never Thought Existed! - Crankk.io

📅 February 27, 2026 👤 scraface

Feb 27, 2026

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📌 c = \sqrt{100} = 10

📌 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

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