Question 108
Let $k$ be a positive integer and let $x_0, x_1, x_2, \dots$ be an infinite sequence defined by the relationship
$$
\begin{aligned}
x_0 &= 0 \\
x_1 &= 1 \\
x_{n+1} &= kx_n + x_{n-1} \quad \text{for all } n \ge 1.
\end{aligned}
$$
a. For the special case $k = 1$, prove that $x_{n-1}x_{n+1}$ is never a perfect square for $n \ge 2$.
b. For the general case of integers $k \ge 1$, prove that $x_{n-1}x_{n+1}$ is never a perfect square for $n \ge 2$.
$$
\begin{aligned}
x_0 &= 0 \\
x_1 &= 1 \\
x_{n+1} &= kx_n + x_{n-1} \quad \text{for all } n \ge 1.
\end{aligned}
$$
a. For the special case $k = 1$, prove that $x_{n-1}x_{n+1}$ is never a perfect square for $n \ge 2$.
b. For the general case of integers $k \ge 1$, prove that $x_{n-1}x_{n+1}$ is never a perfect square for $n \ge 2$.