MathDB
A strange recurrence relation

Source: kjmo 2022 P5

October 29, 2022
recursionalgebra

Problem Statement

A sequence of real numbers a1,a2,…a_1, a_2, \ldots satisfies the following conditions. a1=2a_1 = 2, a2=11a_2 = 11. for all positive integer nn, 2an+2=3an+5(an2+an+12)2a_{n+2} =3a_n + \sqrt{5 (a_n^2+a_{n+1}^2)} Prove that ana_n is a rational number for each of positive integer nn.
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