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Problem 950

AMC 12 late, AIME early

Algebra Difficulty 4.8 Prove it Bulgarian Spring Tournament · Bulgaria

Given is a sequence $a_1, a_2, \dots$ , such that $a_1 = 1$ and $a_{n+1} = \frac{9a_n+4}{a_n+6}$ for any $n \in \mathbb{N}$ . Which terms of this sequence are positive integers?

This one wants a proof. Work it on paper, then read the official solution and mark yourself. Be honest about it: the record is only any use to you if it is.

Official solution

It can be shown by induction that
$a_n = \frac{4 \cdot 2^n - 3}{2^n + 3}, \forall n \ge 2$
Thus, $a_n$ is integer only when $n = 1$ . □

Source: MathNet, licensed CC-BY-4.0. Statement reproduced verbatim; metadata (topic, difficulty, ordering) added by this project.