Question 11
The geometric mean (G.M.) of a $k$ positive numbers $a_{1}, a_{2}, \ldots, a_{k}$ is defined to be the (positive) $k$-th root of their product. For example, the G.M. of $3,4,18$ is $6$. Show that the G.M. of a set $S$ of $n$ positive numbers is equal to the G.M. of the G.M.'s of all non-empty subsets of $S$.