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x_{n+2}=3x_{n+1}-2 x_n , y_n=x^2_n+2^{n+2}, y_n is odd's perfect square

Source: Source: 2019 Nigerian Senior Mathematics Olympiad Round 4 (final) problem 4

September 9, 2019

Perfect SquareoddSequencerecurrence relationalgebranumber theory

Problem Statement

We consider the real sequence (

x_n

) defined by

x_0=0, x_1=1

and

x_{n+2}=3x_{n+1}-2 x_{n}

for

n=0,1,2,...

We define the sequence (

y_n

) by

y_n=x^2_n+2^{n+2}

for every nonnegative integer

n

. Prove that for every

n>0, y_n

is the square of an odd integer.