# Prove the following inequality using the AM-GM inequality. [duplicate]

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