A geometric series has first term a a\,a and common ratio rrr, where r≠1r \neq 1r=1. The sum of the first n n\,n terms of the series is SnS_nSn.
By writing down expressions for Sn S_n\,Sn and for rSn rS_n\,rSn and subtracting one from the other, prove that
Sn=a(1−rn)1−rS_n = \dfrac{a(1 - r^n)}{1 - r}Sn=1−ra(1−rn)