Growth of a Recursion

★ 8/10Recursion sequences Induction Telescoping inequality

Problem

Let $a_1 = 1$ , $a_2 = 8$ , and

a_{n+1} = a_{n-1} + \frac{4}{n}a_n \qquad (n = 2, 3, \ldots).

1. Prove that there exists $c > 0$ with $a_n \leqslant cn^2$ for all $n$ . 2. Prove that $a_{n+1} - a_n \leqslant 4n + 3$ for all $n \in \mathbb{N}^*$ .

Answer

Solution

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

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