Arithmetic-Geometric Progression

Arithmetic-Geometric Progression

## Definition / Concept

An arithmetic-geometric progression is a sequence whose terms contain both an arithmetic and a geometric component.

## Formula

A common form is:

$$a+(a+d)r+(a+2d)r^2+\cdots+(a+(n-1)d)r^{n-1}$$

Its sum is:

$$S_n=a\frac{1-r^n}{1-r}+d\frac{r-nr^n+(n-1)r^{n+1}}{(1-r)^2}$$

for:

$$r\ne1$$

## Usage

- Used to evaluate finite sums in which coefficients form an AP while powers form a GP.