Definition:Arithmetic Sequence

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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-tic.

This is because the word is being used in its adjectival form.