polynomials - Prove $a^n+1$ is divisible by $a + 1$ if $n$ is odd

Prove $a^n+1$ is divisible by $a + 1$ if $n$ is odd:

We know $a$ cannot be $-1$ and the $n \in \mathbb{N}$.
Since $n$ must be odd, we can rewrite $n$ as $2k+1$. Now we assume it holds for prove that it holds for the next term.

$$a^{2(k+1)+1}+1$$
$$=a^{2k+3}+1$$
$$=a^3\cdot a^{2k}+1$$
$$=(a^3+1)\cdot a^{2k} -a^{2k}+1$$

Im not sure on what to do next. Since $a^{2k}$ means that the exponential term will be even and thus you cant use the fact that $a^n+1$ is divisible by $a + 1$ if $n$ is odd.