Proof that for $a>0$ and $a + 1/a$ element of $\mathbb{Z}$, $a^n + 1/a^n$ is always element of $\mathbb{Z}$ by induction

Question

Question is in the title. I'd like to prove that for $a>0$ ($a$ is an element of $\mathbb{R}$) and $a + 1/a$ an element of $\mathbb{Z}$, $a^n + 1/a^n$ is always element of $\mathbb{Z}$ by induction.

Answer 1

Hint:

$$a^{k+1}+\frac{1}{a^{k+1}}=\left(a^{k}+\frac{1}{a^k}\right)\left(a+\frac{1}{a}\right)-\left(a^{k-1}+\frac{1}{a^{k-1}}\right)$$