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x_nx_{n+2} - x^2_{n+1} = 4^{n-1}.

Source: 2015 Saudi Arabia GMO TST II p1

July 26, 2020
floor functionSequencerecurrence relationalgebra

Problem Statement

Let be given the sequence (xn)(x_n) defined by x1=1x_1 = 1 and xn+1=3xn+⌊xn5⌋x_{n+1} = 3x_n + \lfloor x_n \sqrt5 \rfloor for all n=1,2,3,...,n = 1,2,3,..., where ⌊x⌋\lfloor x \rfloor denotes the greatest integer that does not exceed xx. Prove that for any positive integer nn we have xnxn+2−xn+12=4n−1x_nx_{n+2} - x^2_{n+1} = 4^{n-1}
Trần Nam Dũng
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