Recursive/Strong Induction

Question

Suppose that $a_0, a_1, a_2, \dots$ is a sequence defined as follows. $$a_0 = 2, a_1 = 4, a_2 = 6 \text{, and } a_k = 7 a_{k-3} \text{ for all integers $k \ge 3$.}$$ Prove that $a_n$ is even for all integers $n \ge 0$.

Answer 1

Except the basic case is trivial and assuming that the statement holds for each $k \leq n - 1$ you have $a_n = 7a_{n-3} $. By hypothesis $a_{k-3} = 2\gamma$, $\gamma$ integer, so $a_n = 14 \gamma$ which implies that 2 divides $a_n $.