Definition:Arithmetic Series

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

\(\ds S_n\) \(=\) \(\ds \sum_{k \mathop = 0}^{n - 1} a + k d\)
\(\ds \) \(=\) \(\ds a + \paren {a + d} + \paren {a + 2 d} + \cdots + \paren {a + \paren {n - 1} d}\)

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.