TITLE: Using strong induction to get the AM-GM inequality for $2^n$ numbers
QUESTION [5 upvotes]: The arithmetic mean of $k$ numbers $a_1, a_2, \ldots, a_k$ is their average $\frac{a_1+a_2+\cdots+a_k}{k}=AM$. Their geometric mean
is $\sqrt[k]{a_1a_2\cdots a_k}=GM$. I am asked to show this:

Use induction to prove: If $k=2^n$ and if all the numbers $a_1, a_2, \ldots, a_k$ are nonnegative, then $AM \geq GM$. 

I'll be honest, I have no work for this problem.  I've looked into many examples of strong induction, but most of them are abstract.  Please give me insight for what method to best utilize for strong induction.

REPLY [2 votes]: The use of induction (first ordinary induction to prove the result for powers of $2$, then upward-downward induction for the general case) to prove the Arithmetic-Geometric Mean Inequality appears in my lecture notes on induction (Theorem 21 in $\S 12$).
I agree that this argument is somewhat tricky.  I am currently teaching a "Spivak calculus course" (i.e., very strong, very committed university freshmen) and I put this argument into my notes (which originated in a different undergraduate course) expressly for the current course but have not, so far, had a chance to mention it in class.  I think many if not most young students would have trouble supplying a complete proof.