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Home » A sequence is defined by \(a_n = 2n + 3\). Find the 10th term. - Simpleprint

A sequence is defined by \(a_n = 2n + 3\). Find the 10th term. - Simpleprint

📅 March 14, 2026 👤 scraface

Mar 14, 2026

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📌 Question: An epidemiologist models the spread of a disease with the polynomial $ g(x) $, where $ g(x^2 - 1) = 2x^4 - 5x^2 + 1 $. Find $ g(x^2 + 1) $.

📌 Solution: Let $ y = x^2 - 1 $. Then $ x^2 = y + 1 $. Substitute into $ g(y) = 2x^4 - 5x^2 + 1 $:

📌 x^4 = (x^2)^2 = (y + 1)^2 = y^2 + 2y + 1.

📌 g(y) = 2(y^2 + 2y + 1) - 5(y + 1) + 1 = 2y^2 + 4y + 2 - 5y - 5 + 1 = 2y^2 - y - 2.

📌 Now, substitute $ y = x^2 + 1 $ to find $ g(x^2 + 1) $:

📌 g(x^2 + 1) = 2(x^2 + 1)^2 - (x^2 + 1) - 2 = 2x^4 + 4x^2 + 2 - x^2 - 1 - 2 = 2x^4 + 3x^2 - 1.

📌 The final answer is $ \boxed{2x^4 + 3x^2 - 1} $.

📌 Question: An angel investor analyzes a medical startup's growth, modeled by the polynomial $ g(x) $, where $ g(x^2 + 2) = 3x^6 - 4x^4 + 5x^2 - 2 $. Find $ g(x^2 - 2) $.

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