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Home » our deepest fear is not that we are inadequate - Abbey Badges

our deepest fear is not that we are inadequate - Abbey Badges

📅 February 26, 2026 👤 scraface

Feb 26, 2026

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📌 f(u) = u^2 - 1.

📌 Now substitute $ u = x^2 - 2 $ into $ f $:

📌 f(x^2 - 2) = (x^2 - 2)^2 - 1 = x^4 - 4x^2 + 4 - 1 = x^4 - 4x^2 + 3.

📌 \boxed{x^4 - 4x^2 + 3}

📌 Question: Compute $ \sum_{n=1}^{50} \frac{1}{n(n+2)} $.

📌 Solution: We begin by simplifying the general term using partial fractions:

📌 \frac{1}{n(n+2)} = \frac{A}{n} + \frac{B}{n+2}.

📌 Multiplying both sides by $ n(n+2) $:

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