A Decreasing Recursion

★ 7/10Recursion sequences Induction Telescoping inequality

Problem

A sequence has $a_1 = \dfrac{1}{2}$ and

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

and $b_n = na_n$ .

1. Prove that $\{a_n\}$ is decreasing. 2. Prove that $b_n \leqslant \dfrac{3}{2}$ for all $n \in \mathbb{N}^*$ .

Answer

Solution

Difficulty	7/10
Topics	Recursion, sequences, Induction, Telescoping, inequality

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