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Home » Stop Spelling 'Miscellaneous' Wrong—Here’s How! The Simple Solution! - American Beagle Club

Stop Spelling 'Miscellaneous' Wrong—Here’s How! The Simple Solution! - American Beagle Club

📅 February 27, 2026 👤 scraface

Feb 27, 2026

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a^3 + b^3 + c^3 - 3abc = (a + b + c)(a^2 + b^2 + c^2 - ab - bc - ca).
Given that $ a + b + c = 0 $, substitute into the identity:
a^3 + b^3 + c^3 - 3abc = 0 \cdot (a^2 + b^2 + c^2 - ab - bc - ca) = 0.
a^3 + b^3 + c^3 = 3abc.
This completes the proof.
oxed{a^3 + b^3 + c^3 = 3abc}
Question: Compute $ \sum_{k=1}^{50} rac{1}{k(k+2)} $.
Solution: We evaluate the sum:
\sum_{k=1}^{50} rac{1}{k(k+2)}.
We use partial fractions:

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