Question 10
Sequences $(x_n)$ and $(y_n)$ are given recursively:
$x_1 = 3$, $y_1 = 1$,
$x_{n+1} = 3x_n + y_n$, for all $n \in \mathbb{N}$;
$y_{n+1} = x_n + 3y_n$, for all $n \in \mathbb{N}$.
Prove that $x_{2017}^2 - y_{2017}^2 = 8^{2017}$.
$x_1 = 3$, $y_1 = 1$,
$x_{n+1} = 3x_n + y_n$, for all $n \in \mathbb{N}$;
$y_{n+1} = x_n + 3y_n$, for all $n \in \mathbb{N}$.
Prove that $x_{2017}^2 - y_{2017}^2 = 8^{2017}$.