# Definition:Arithmetic-Geometric Sequence

## Definition

An arithmetic-geometric sequence is a sequence $\sequence {a_k}$ in $\R$ defined as:

$a_k = \paren {a_0 + k d} r^k$

for $k = 0, 1, 2, \ldots$

Thus its general form is:

$a_0, \paren {a_0 + d} r, \paren {a_0 + 2 d} r^2, \paren {a_0 + 3 d} r^3, \ldots$

## Also known as

An arithmetic-geometric sequence is also (more usually) known as an arithmetic-geometric progression.

$\mathsf{Pr} \infty \mathsf{fWiki}$ prefers the word sequence because it is less ambiguous than progression, which is sometimes also seen to mean series.

## Also see

• Results about Arithmetic-Geometric 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.