Question 25
A sequence of integers $a_{1}, a_{2}, \ldots$, is such that $a_{1}=1$, $a_{2}=2$ and for $n \geq 1$
$$
a_{n+2}= \begin{cases}5 a_{n+1}-3 a_{n} & \text{ if } a_{n} \cdot a_{n+1} \text{ is even, } \\ a_{n+1}-a_{n} & \text{ if } a_{n} \cdot a_{n+1} \text{ is odd. }\end{cases}
$$
Prove that $a_{n} \neq 0$ for all $n$.
$$
a_{n+2}= \begin{cases}5 a_{n+1}-3 a_{n} & \text{ if } a_{n} \cdot a_{n+1} \text{ is even, } \\ a_{n+1}-a_{n} & \text{ if } a_{n} \cdot a_{n+1} \text{ is odd. }\end{cases}
$$
Prove that $a_{n} \neq 0$ for all $n$.