2015 OIM Problems/Problem 3

Problem

Let $\alpha$ and $\beta$ be roots of the polynomial $x^2-qx+1$ , where $q$ is a rational number greater than 2. We define $s_1=\alpha+\beta$ , $t_1=1$ , and for each integer $n \ge 2$ ,

$s_n=\alpha ^n +\beta ^n,\;t_n=s_{n-1}+2s_{n-2}+\cdots+(n-1)s_1+n$

Show that for all odd $n$ , $t_n$ is the square of a rational number.

~translated into English by Tomas Diaz. ~orders@tomasdiaz.com

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2015 OIM Problems/Problem 3

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