Problem

Source: 2018 China North Mathematical Olympiad Grade 11 Test 2 P2

Tags: number theory, Sequence



For $a_1 = 3$, define the sequence $a_1, a_2, a_3, \ldots$ for $n \geq 1$ as $$na_{n+1}=2(n+1)a_n-n-2.$$Prove that for any odd prime $p$, there exist positive integer $m,$ such that $p|a_m$ and $p|a_{m+1}.$