Prove the following inequality using the AM-GM inequality.

Question

I want to be able to prove that for any positive real numbers $a_1, ... , a_n$, that $$(1 + a_1)\cdots(1+a_n) \ge 2^n \sqrt{a_1\cdots a_n}$$

I know that I must use the AM-GM inequality in some way, i.e. I want to use the fact that $a_1 + a_2 + \cdots + a_n \ge n*(a_1a_2\cdots a_n)^{1/n}$, but I am not sure where to start. I also was considering using induction, but it isn't immediately clear what the base case is or if induction is even possible. Any guidance would be appreciated greatly.

Answer 1

Just repeatedly use the $n=2$ case of the AM-GM inequality: $1+a_i\geq 2\sqrt{a_i}$ for $1\leq i\leq n$.