Show that $a_n$ is divisible by 4 for all integers $n \geq 1.$

Question

Define the sequence $a_1, a_2, ...$ recursively by:\begin{align*}a_1 &= 4, \\a_2 &= 8, \\a_k & = 5a_{k-1} - a_{k-2}^2 \ \ \ \ \ \ \ \text{ for } k \geq 3.\end{align*}Show that $a_n$ is divisible by $4$ for all integers $n \geq 1$.

Answer 1

Hint: Use strong induction. Assume the statement is true for all $i$, such that $k\geq i\geq3$. And prove that the statement is true for $k+1$. So We know that $4|a_i$ for all $k\geq i$. Now $a_{k+1}=5a_k-a_{k-1}^2$. We know that $4|a_k$ and $4|a_{k-1}$. Can you take it form here?