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determine n such that terms of a_n are perfect squares

Source: Balkan MO ShortList 2008 N6

April 5, 2020

Problem Statement

Let (xn)(x_n), n=1,2,…n=1,2, \ldots be a sequence defined by x1=2008x_1=2008 and \begin{align*} x_1 +x_2 + \ldots + x_{n-1} = \left( n^2-1 \right) x_n \qquad ~ ~ ~ \forall n \geq 2 \end{align*} Let the sequence an=xn+1nSna_n=x_n + \frac{1}{n} S_n, n=1,2,3,…n=1,2,3, \ldots where SnS_n == x1+x2+…+xnx_1+x_2 +\ldots +x_n. Determine the values of nn for which the terms of the sequence ana_n are perfect squares of an integer.
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