# Definition:Arithmetic-Geometric Progression

## Definition

The term arithmetic-geometric progression is used to mean one of the following:

### Arithmetic-Geometric Sequence

An arithmetic-geometric sequence is a sequence $\sequence {a_k}$ in $\R$ defined as:

$a_k = \paren {a_0 + k d} r^k$

for $k = 0, 1, 2, \ldots$

Thus its general form is:

$a_0, \paren {a_0 + d} r, \paren {a_0 + 2 d} r^2, \paren {a_0 + 3 d} r^3, \ldots$

### Arithmetic-Geometric Series

An arithmetic-geometric series is a series whose underlying sequence is an arithmetic-geometric sequence:

 $\displaystyle S_n$ $=$ $\displaystyle \sum_{k \mathop = 0}^{n - 1} \paren {a + k d} r^k$ $\displaystyle$ $=$ $\displaystyle a + \paren {a + d} r + \paren {a + 2 d} r^2 + \cdots + \paren {a + \paren {n - 1} d}r^{n-1}$