# Definition:Arithmetic Sequence

## Definition

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:

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

### Initial Term

The term $a_0$ is the **initial term** of $\sequence {a_k}$.

### Common Difference

The term $d$ is the **common difference** of $\sequence {a_k}$.

### Last Term

The term $a_{n-1} = a_0 + \paren {n - 1} d$ is the **last term** of $\sequence {a_k}$.

## Also known as

The term **arithmetic progression** is usual.

**Arithmetical progression** is also sometimes seen.

Hence the abbreviation **A.P.** is well-understood.

However, use of the term **progression**, although ubiquitous in the literature, is deprecated on $\mathsf{Pr} \infty \mathsf{fWiki}$ on account of the fact that the term is used both for **arithmetic sequence** and **arithmetic series**, and this can be a source of confusion.

## Also see

- Results about
**arithmetic sequences**can be found**here**.

## Linguistic Note

In the context of an **arithmetic sequence** or **arithmetic-geometric sequence**, 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.

## Sources

- 1989: Ephraim J. Borowski and Jonathan M. Borwein:
*Dictionary of Mathematics*... (previous) ... (next):**arithmetic progression** - 1998: David Nelson:
*The Penguin Dictionary of Mathematics*(2nd ed.) ... (previous) ... (next):**arithmetic progression (arithmetic sequence)** - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next):**arithmetic progression (arithmetic sequence)** - 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next):**arithmetic sequence** - 2021: Richard Earl and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(6th ed.) ... (previous) ... (next):**arithmetic sequence**