Saudi Arabia Mathematical Competitions 2012

Question

Sequence (x_1, x_2, ) is defined as x_1 = 20, x_2 = 12, x_n+2 = x_n + x_n+1 + 2x_n x_n+1 + 121, for n 1. 1) Compute x_10. 2) Determine with justification if every term in the sequence is an integer?

Accepted Answer

It is clear that x_3 = 20 + 12 + 2 19 = 70. We note that aligned x_n+3 &= x_n+1 + x_n+2 + 2x_n+1x_n+2 + 121 &= x_n+1 + x_n+2 + 2x_n+1(x_n + x_n+1 + 2x_n x_n+1 + 121) + 121 &= x_n+1 + x_n+2 + 2x_n+1^2 + 2x_n+1x_n x_n+1 + 121 + x_n x_n+1 + 121 &= x_n+1 + x_n+2 + 2(x_n+1 + x_n x_n+1 + 121) &= 3x_n+1 + x_n+2 + 2x_n x_n+1 + 121 &= 3x_n+1 + x_n+2 + x_n+2 - x_n - x_n+1 &= 2x_n+2 + 2x_n+1 - x_n. aligned Therefore, (x_1, x_2, ) is an integer sequence and it is straightforward to find x_4 = 144, x_5 = 416, x_6 = 1050, x_7 = 2788, x_8 = 7260, x_9 = 19046, and x_10 = 49824.