Four Claims About a Rational Iteration

★ 8/10Monotonicity Recursion sequences Casework

Problem

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

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

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

1. If $a<1$ , then $a_n<1$ for all $n$ . 2. For every $a\in\mathbb R$ , $0\leqslant a_{n+1}<2$ for all $n$ . 3. If $0<a<1$ , then $a_{n+1}<a_n$ for all $n$ . 4. For every positive integer $n$ , $(a-1)(a_n-1)\geqslant 0$ .

Answer

Solution

Difficulty	8/10
Topics	Monotonicity, Recursion, sequences, Casework

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