# Definition:Arithmetic Series

## Definition

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

 $\displaystyle S_n$ $=$ $\displaystyle \sum_{k \mathop = 0}^{n - 1} a + k d$ $\displaystyle$ $=$ $\displaystyle a + \paren {a + d} + \paren {a + 2 d} + \cdots + \paren {a + \paren {n - 1} d}$

## 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.