Bounding a Quadratic Recursion

★ 8/10Recursion sequences Estimation Telescoping inequality

Problem

A sequence $\{a_n\}$ has $a_n > 1$ , $a_1 = 2$ , and

a_{n+1}^2 - a_{n+1} - a_n^2 + 1 = 0.

1. Prove that $\dfrac{n+7}{4} \leqslant a_n < a_{n+1} \leqslant n + 2$ . 2. Prove that $\displaystyle\sum_{k=1}^n \frac{1}{2a_k^2 - 3} < 1$ .

Answer

Solution

Difficulty	8/10
Topics	Recursion, sequences, Estimation, Telescoping, inequality

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