Squeezed Between a Half and One

★ 8/10sequences Induction Telescoping inequality

Problem

A sequence has $a_1 = \dfrac13$ and

a_{n+1} = a_n + \frac{a_n^2}{n^2}.

1. Prove $a_n < a_{n+1} < 1$ for all $n$ . 2. Prove $a_n > \dfrac12 - \dfrac{1}{4n}$ for all $n$ .

Answer

Solution

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

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