# Definition:Arithmetic Series

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## Definition

An **arithmetic series** is a series whose underlying sequence is an arithmetic progression:

\(\displaystyle S_n\) | \(=\) | \(\displaystyle \sum_{k \mathop = 0}^{n - 1} a + k d\) | |||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle a + \left({a + d}\right) + \left({a + 2 d}\right) + \cdots + \left({a + \left({n - 1}\right) d}\right)\) |

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

## Sources

- 1968: Murray R. Spiegel:
*Mathematical Handbook of Formulas and Tables*... (previous) ... (next): $\S 19$: Arithmetic Series: $19.1$