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Home » The \( n \)-th term is \( a_n = ar^n-1 \). - Imagemakers

The \( n \)-th term is \( a_n = ar^n-1 \). - Imagemakers

📅 April 30, 2026 👤 scraface

Apr 30, 2026

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📌 So $ R(x) = (x - 1)(x^2 + x - 2) $. Factor the quadratic:

📌 x^2 + x - 2 = (x + 2)(x - 1).

📌 R(x) = (x - 1)^2(x + 2).

📌 The roots are $ x = 1 $ (with multiplicity 2) and $ x = -2 $. Therefore, there are **two distinct real roots**, but **three real roots counting multiplicity**. Since the question asks for the number of real roots (not distinct), the answer is:

📌 Question: A zoologist studying social behavior models the interaction between two animals with the function $ I(u) = u - \frac{u^3}{9} $. If $ n $ is a positive integer, define $ x_n $ as the value of $ u $ such that $ I(x_n) = \frac{1}{n} $. Find $ \lim_{n \to \infty} x_n $.

📌 Solution: We are given $ I(u) = u - \frac{u^3}{9} $, and $ I(x_n) = \frac{1}{n} $. So:

📌 x_n - \frac{x_n^3}{9} = \frac{1}{n}.

📌 We want to evaluate $ \lim_{n \to \infty} x_n $. As $ n \to \infty $, $ \frac{1}{n} \to 0 $, so $ x_n \to 0 $. Let’s refine the approximation. Assume $ x_n \approx \frac{c}{n} $ for some constant $ c $. Then

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