Question 121
The sequences $\{x_{n}\}$ and $\{y_{n}\}$ are defined by $x_{0}=2$, $y_{0}=1$ and, for $n \geq 0$,
$$
x_{n+1}=x_{n}^{2}+y_{n}^{2} \quad \text{ and } \quad y_{n+1}=2 x_{n} y_{n}
$$
Find and prove an explicit formula for $x_{n}$ in terms of $n$.
$$
x_{n+1}=x_{n}^{2}+y_{n}^{2} \quad \text{ and } \quad y_{n+1}=2 x_{n} y_{n}
$$
Find and prove an explicit formula for $x_{n}$ in terms of $n$.