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\boxed\left(\frac43, \frac173\right) - Groen Casting

📅 March 1, 2026 👤 scraface

Mar 01, 2026

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📌 Insect Type Pokémon Breaking Records – Here’s Their Mind-Blowing Genius!Question: A sequence of four real numbers forms an arithmetic progression with common difference $ d $. If the sum of the squares of the first and last terms equals the square of the sum of all four terms, find the ratio of the third term to the first term.

📌 Solution: Let the four terms be $ a - \frac{3d}{2},\ a - \frac{d}{2},\ a + \frac{d}{2},\ a + \frac{3d}{2} $. This ensures the terms are in arithmetic progression with common difference $ d $. The first term is $ a - \frac{3d}{2} $, and the last term is $ a + \frac{3d}{2} $. Their sum of squares is:

📌 \left(a - \frac{3d}{2}\right)^2 + \left(a + \frac{3d}{2}\right)^2 = 2a^2 + 2\left(\frac{9d^2}{4}\right) = 2a^2 + \frac{9d^2}{2}.

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