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A sequence of reals which produce perfect squares

Source: Belarus TST 2024

July 17, 2024
Sequencealgebranumber theory

Problem Statement

A sequence {yi}\{y_i\} is given, where y0=−14,y1=0y_0=-\frac{1}{4},y_1=0. For every positive integer nn the following equality holds: yn−1+yn+1=4yn+1y_{n-1}+y_{n+1}=4y_n+1 Prove that for every positive integer nn the number 2y2n+322y_{2n}+\frac{3}{2} a) is a positive integer b) is a square of a positive integer D. Zmiaikou
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