An Invariant in a Recursion

★ 8/10Recursion sequences Induction Telescoping inequality

Problem

Sequences $\{a_n\}$ and $\{b_n\}$ satisfy $a_1 = 1$ , $b_1 = 2$ , and

a_{n+1}b_n = a_nb_n + 2a_n + 4.

1. If additionally $b_n = 2a_n$ for all $n$ , prove that for $n \geqslant 2$ ,

n + 2 \leqslant a_n \leqslant \frac{3}{2}n + 1.

If instead $a_nb_{n+1} = a_nb_n + 2b_n + 4$ for all $n$ , prove that $a_n < 10$ for all $n$ .

Answer

Solution

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

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