A Fractional Recursion Bound

★ 8/10Recursion sequences Induction AM-GM inequality

Problem

Let $b > 0$ , and let the sequence $\{a_n\}$ satisfy $a_1 = b$ and

a_n = \frac{nba_{n-1}}{a_{n-1} + 2n - 2} \qquad (n \geqslant 2).

1. Find a formula for $a_n$ . 2. Prove that for every positive integer $n$ ,

a_n \leqslant \frac{b^{n+1}}{2^{n+1}} + 1.

Answer

Solution

Difficulty	8/10
Topics	Recursion, sequences, Induction, AM-GM, inequality

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