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something blue borrowed new old - Midis

📅 March 11, 2026 👤 scraface

Mar 11, 2026

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a^4 r^6 = 20^4 \Rightarrow (ar^{1.5})^4 = 20^4 \Rightarrow ar^{1.5} =...
Letâs write $ a = rac{20}{r^{1.5}} $. Substitute into the sum:
rac{20}{r^{1.5}} (1 + r + r^2 + r^3) = 120
\Rightarrow rac{20(1 + r + r^2 + r^3)}{r^{1.5}} = 120
\Rightarrow rac{1 + r + r^2 + r^3}{r^{1.5}} = 6
Try $ r = 1 $:
rac{1 + 1 + 1 + 1}{1} = 4 < 6
Try $ r = 2 $:
rac{1 + 2 + 4 + 8}{2^{1.5}} = rac{15}{2\sqrt{2}} pprox rac{15}{2.82...
Try $ r = 3 $:

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