Definition:Arithmetic Progression

Definition

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

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:

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

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 $

$=$

$\ds a + \paren {a + d} + \paren {a + 2 d} + \cdots + \paren {a + \paren {n - 1} d}$