Bounded Yet Exceeding Four

★ 9/10Recursion sequences Estimation Telescoping inequality

Problem

A sequence $\{a_n\}$ satisfies $a_1 = 1$ and

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

1. Prove that $a_n < 5$ for every positive integer $n$ . 2. Prove that there is a positive integer $m$ with $a_m > 4$ .

Answer

Solution

Difficulty	9/10
Topics	Recursion, sequences, Estimation, Telescoping, inequality

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