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Eventually constant

Source: China South East Mathematical Olympiad 2016 Grade 10 Prob. 8

July 31, 2016
number theoryalgebra

Problem Statement

Let {an}\{ a_n\} be a series consisting of positive integers such that n2∣∑i=1nain^2 \mid \sum_{i=1}^{n}{a_i} and an≤(n+2016)2a_n\leq (n+2016)^2 for all n≥2016n\geq 2016. Define bn=an+1−anb_n=a_{n+1}-a_n. Prove that the series {bn}\{ b_n\} is eventually constant.
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