MathDB

4

Part of 2019 Nigerian Senior MO Round 4

Problems(1)

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

9/9/2019
We consider the real sequence (xnx_n) defined by x0=0,x1=1x_0=0, x_1=1 and xn+2=3xn+1−2xnx_{n+2}=3x_{n+1}-2 x_{n} for n=0,1,2,...n=0,1,2,... We define the sequence (yny_n) by yn=xn2+2n+2y_n=x^2_n+2^{n+2} for every nonnegative integer nn. Prove that for every n>0,ynn>0, y_n is the square of an odd integer.
Perfect SquareoddSequencerecurrence relationalgebranumber theory