# Arithmetic-Geometric Series The sum of first $n$ terms of an [[Arithmetic-Geometric Sequence]] can be calculated as: $\sum_{k=1}^n [a+(k-1)d]r^{k-1} = -\frac{a-[a+(n-1)d]r^n}{r-1} - \frac{r^{n-1}-1}{(r-1)^2}dr$ ^3peyiy For a sum of infinite terms see [[Arithmetic-Geometric Infinite Series]]. ## Proof Proof is similar to [[Geometric Sequence#Geometric Series#Proof]]. After subtracting the $S_n$ from $rS_n$, you should realize that all terms of the new sequence, except for the first and last terms, form a geometric sequence themselves. For details see [Arithmetic-Geometric Progression](https://brilliant.org/wiki/arithmetic-geometric-progression/) from Brilliant.