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sequence is sum of consecutive squares

Source: Yugoslav TST 1996 P3

May 16, 2021
number theorySequences

Problem Statement

The sequence {xn}\{x_n\} is given by xn=14((2+3)2n−1+(2−3)2n−1),n∈N.x_n=\frac14\left(\left(2+\sqrt3\right)^{2n-1}+\left(2-\sqrt3\right)^{2n-1}\right),\qquad n\in\mathbb N.Prove that each xnx_n is equal to the sum of squares of two consecutive integers.
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