# Definition:Arithmetic Progression

## Definition

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

### Arithmetic Sequence

An arithmetic sequence is a finite sequence $\sequence {a_k}$ in $\R$ or $\C$ defined as:

$a_k = a_0 + k d$ for $k = 0, 1, 2, \ldots, n - 1$

Thus its general form is:

$a_0, a_0 + d, a_0 + 2 d, a_0 + 3 d, \ldots, a_0 + \paren {n - 1} d$

### Arithmetic Series

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

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