IrMO 2010 Problem 9

Let $n\ge 3$ be an integer and $a_1,a_2,\dots ,a_n$ be a finite sequence of positive integers, such that, for $k=2,3,\dots ,n$ $$n(a_k+1)-(n-1)a_{k-1}=1.$$ Prove that $a_n$ is not divisible by $(n-1)^2$ .