Definition:Arithmetic Progression

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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-tic.

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