<s>[INST] Generalizing proof of AM-GM inequality? For two positive real numbers $x$ and $y$, we can prove the AM-GM inequality between them.
$$\left(\sqrt{x}-\sqrt{y}\right)^2=x+y-2\sqrt{xy}\geq 0$$
$$\implies \frac{x+y}{2}\geq\sqrt{xy}$$
Can this be generalized easily to the case of more numbers?
 [/INST] If $n=2^k$, where $k\in\mathbb N$, so we can use $\frac{x+y}{2}\geq\sqrt{xy}$ for non-negatives $x$ and $y$.
If $n>2^k$ we can use a similar to the following reasoning.
We'll prove that $\frac{x+y+z}{3}\geq\sqrt[3]{xyz}$ for non-negatives $x$, $y$ and $z$.
Indeed, by AM-GM for four variables we obtain:
$$\frac{x+y+z+\sqrt[3]{xyz}}{4}\geq\sqrt[4]{xyz\sqrt[3]{xyz}}=\sqrt[3]{xyz}$$
and we are done!
For five non-negative variables we have
$$\frac{a+b+c+d+e+3\sqrt[5]{abcde}}{8}\geq\sqrt[8]{abcde\sqrt[5]{a^3b^3c^3d^3e^3}}=\sqrt[5]{abcde}$$
which gives
$$\frac{a+b+c+d+e}{5}\geq\sqrt[5]{abcde}$$
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