# Definition:Arithmetic Progression

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## Contents

## Definition

An **arithmetic progression** is a finite sequence $\left \langle{a_k}\right \rangle$ 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:

- $a_0, a_0 + d, a_0 + 2 d, a_0 + 3 d, \ldots, a_0 + \left({n - 1}\right) d$

### Initial Term

The term $a_0$ is the **initial term** of $\left \langle{a_k}\right \rangle$.

### Common Difference

The term $d$ is the **common difference** of $\left \langle{a_k}\right \rangle$.

### Last Term

The term $a_{n-1} = a_0 + \left({n - 1}\right) d$ is the **last term** of $\left \langle{a_k}\right \rangle$.

## Also known as

The term **arithmetical progression** is sometimes seen.

## Also see

- Results about
**Arithmetic Progressions**can be found here.

## Linguistic Note

In the context of an **arithmetic progression** or **arithmetic-geometric progression**, the word **arithmetic** is pronounced with the stress on the first and third syllables: ** a-rith-me-tic**, rather than on the second syllable:

**a-**.

*rith*-me-ticThis is because the word is being used in its adjectival form.