The idea of the induction using can be the following.
For $n=2^k$, where $k$ is a natural number we can use AM-GM number times, like for $n=4$:
For non-negatives $a$, $b$, $c$ and $d$ we obtain:
$$\frac{a+b+c+d}{4}\geq\frac{2\sqrt{ab}+2\sqrt{cd}}{4}=\frac{\sqrt{ab}+\sqrt{cd}}{2}\geq\sqrt{\sqrt{ab}\cdot\sqrt{cd}}=\sqrt[4]{abcd}.$$
Now, for $n=3$ we obtain:
$$\frac{a+b+c+\sqrt[3]{abc}}{4}\geq\sqrt[4]{abc\sqrt[3]{abc}}=\sqrt[3]{abc},$$
which gives $$\frac{a+b+c}{3}\geq\sqrt[3]{abc}.$$
Or for non-negatives $a$, $b$, $c$, $d$ and $e$ we obtain:
$$\frac{a+b+c+d+e+3\sqrt[5]{abcde}}{8}\geq\sqrt[8]{abcde\left(\sqrt[5]{abcde}\right)^3}=\sqrt[5]{abcde},$$
which gives
$$\frac{a+b+c+d+e}{5}\geq\sqrt[5]{abcde}.$$