Question 72
The progression $\{x_n\}$ is defined by the first two terms $x_1 = a$ and $x_2 = b$ and a recursive formula
$$
x_n = 3x_{n-1} - x_{n-2}
$$
for $n \ge 3$. Prove that there exist positive integers $a$ and $b$, such that the number $1 + x_n x_{n+1}$ is a perfect square for all $n$.
$$
x_n = 3x_{n-1} - x_{n-2}
$$
for $n \ge 3$. Prove that there exist positive integers $a$ and $b$, such that the number $1 + x_n x_{n+1}$ is a perfect square for all $n$.