G.P.: first term (a), common ratio (r)

[ S_n = a\dfrac{1-r^n}{1-r} ]

Sum from ((n+1)^{th}) to ((2n)^{th}): [ S' = a_{n+1}+a_{n+2}+\cdots+a_{2n} = ar^n\dfrac{1-r^n}{1-r} ]

Ratio: [ \dfrac{S_n}{S'} = \dfrac{a\dfrac{1-r^n}{1-r}}{ar^n\dfrac{1-r^n}{1-r}} = \dfrac{1}{r^n} ]

Proved.