Squeezed by a Recurrence

Problem

A sequence $\{a_n\}$ satisfies $a_1 = 3$ and

a_n^2 - (1 + a_{n+1})\,a_n + 2 = 0 \qquad (n \in \mathbb{N}^*).

1. Prove that $2 < a_{n+1} < a_n$ for all $n$ . 2. Let $S_n = a_1 + a_2 + \cdots + a_n$ . Prove that

2 - 2\left(\frac12\right)^n \;\le\; S_n - 2n \;\le\; 3 - 3\left(\frac23\right)^n.

Answer

Solution

Difficulty	7/10
Topics	sequences, inequality

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