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Home » = 2 \times 3^2 \times 5 - Red Crowns

= 2 \times 3^2 \times 5 - Red Crowns

📅 March 11, 2026 👤 scraface

Mar 11, 2026

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📌 From the product:

📌 a^4 r^6 = 160000 = 16 \cdot 10000 = 16 \cdot 10^4 = (2^4)(10^4) = (2 \cdot 10)^{4} = 20^4

📌 a^4 r^6 = 20^4 \Rightarrow (ar^{1.5})^4 = 20^4 \Rightarrow ar^{1.5} = 20

📌 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 $:

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