Competition problem — Algebra — Maths Olympiad Prep

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Algebra Difficulty 4.8 AIME Prove it 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?

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 $n = 1$ . □

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