# Definition:Arithmetic-Geometric Sequence

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## 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-ticThis is because the word is being used in its adjectival form.