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Recursive Sequence Divisibility

Source: 2010 IrMO Paper 2 Problem 4

January 30, 2018
combinatoricsnumber theory

Problem Statement

Let n≥3n\ge 3 be an integer and a1,a2,…,ana_1,a_2,\dots ,a_n be a finite sequence of positive integers, such that, for k=2,3,…,nk=2,3,\dots ,n n(ak+1)−(n−1)ak−1=1.n(a_k+1)-(n-1)a_{k-1}=1. Prove that ana_n is not divisible by (n−1)2(n-1)^2.
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