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Home » From Calm Skies to Chaos! What the Falling Skies Reveal About Our Planet - Upplift

From Calm Skies to Chaos! What the Falling Skies Reveal About Our Planet - Upplift

📅 March 2, 2026 👤 scraface

Mar 02, 2026

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📌 Substitute $ x = \sqrt{\frac{28}{9}} = \frac{2\sqrt{7}}{3} $:

📌 r = \frac{1 + \frac{1}{2} \cdot \frac{2\sqrt{7}}{3}}{1 - \frac{3}{2} \cdot \frac{2\sqrt{7}}{3}} = \frac{1 + \frac{\sqrt{7}}{3}}{1 - \sqrt{7}}.

📌 As before, this evaluates to $ -\frac{5 + 2\sqrt{7}}{9} $, but this is negative, which contradicts the premise unless the terms are not ordered. However, the ratio of third to first term is geometric in symmetric AP.

📌 Alternatively, let the first term be $ A $, common difference $ d $, so terms: $ A, A+d, A+2d, A+3d $. Sum of squares of first and last:

📌 A^2 + (A+3d)^2 = A^2 + A^2 + 6Ad + 9d^2 = 2A^2 + 6Ad + 9d^2.

📌 Sum of all terms: $ 4A + 6d $, square: $ 16A^2 + 48Ad + 36d^2 $.

📌 Set equal:

📌 2A^2 + 6Ad + 9d^2 = 16A^2 + 48Ad + 36d^2

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