Four Claims About a Recurrence

★ 8/10Mean Value Theorem Monotonicity Recursion sequences inequality

Problem

A sequence $\{a_n\}$ satisfies $a_n>0$ and

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

Determine, with proof, which of the following claims are true.

1. If $a_1=\dfrac 12$ , then $a_n>a_{n+1}$ for all $n$ . 2. If $a_n<a_{n+1}$ for all $n$ , then $a_1>1$ . 3. $a_1+a_5\leqslant 2a_3$ . 4. $|a_{n+2}-a_{n+1}|\leqslant \dfrac{\sqrt 2}2|a_{n+1}-a_n|$ for all $n$ .

Answer

Solution

Difficulty	8/10
Topics	Mean Value Theorem, Monotonicity, Recursion, sequences, inequality

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