Bounding a Logistic Sum

★ 8/10sequences Induction Telescoping inequality

Problem

A sequence $\{a_n\}$ satisfies $a_1=\dfrac 12$ and

a_{n+1}=a_n-a_n^2\qquad(n\in\mathbb N^{*}).

1. Prove that $1\leqslant\dfrac{a_n}{a_{n+1}}\leqslant 2$ for all $n\in\mathbb N^{*}$ . 2. Let $S_n$ be the sum of the first $n$ terms of $\{a_n^2\}$ . Prove that

\frac 1{2(n+2)}\leqslant\frac{S_n}n\leqslant\frac 1{2(n+1)}\qquad(n\in\mathbb N^{*}).

Answer

Solution

Difficulty	8/10
Topics	sequences, Induction, Telescoping, inequality

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