A Logistic-Style Iteration

Problem

A sequence satisfies $2a_{n+1} = 1 - a_n^2$ with $0 < a_1 < 1$ .

1. Find the range of $a_3$ . 2. Prove by induction that for $n \ge 3$ ,

\left|a_n - \left(\sqrt2 - 1\right)\right| < \frac{1}{2^n}.

Let $b_n = \dfrac{1}{a_n}$ . Prove a corresponding estimate for $b_n - \left(\sqrt2 + 1\right)$ when $n \ge 3$ .

Answer

Solution

Difficulty	8/10
Topics	sequences, Estimation, Induction

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