sum(ar^(k-1), k=1 to n) = a$\frac{r^n-1}{(r-1)}$ for r != 1. Infinite GP (|r|<1): S = $\frac{a}{1-r}$. These are not induction topics per se but combine with induction in mixed problems. sum(k*$r^k$) can be evaluated using differentiation of GP sum formula or by the S - rS technique.