Definition:Arithmetic Sequence
Definition
An arithmetic sequence is a finite sequence $\sequence {a_k}$ in $\R$ or $\C$ defined as:
- $a_k = a_0 + k d$ for $k = 0, 1, 2, \ldots, n - 1$
Thus its general form is:
- $a_0, a_0 + d, a_0 + 2 d, a_0 + 3 d, \ldots, a_0 + \paren {n - 1} d$
Initial Term
The term $a_0$ is the initial term of $\sequence {a_k}$.
Common Difference
The term $d$ is the common difference of $\sequence {a_k}$.
Last Term
The term $a_{n-1} = a_0 + \paren {n - 1} d$ is the last term of $\sequence {a_k}$.
Also known as
The term arithmetic progression is usual.
Arithmetical progression is also sometimes seen.
Hence the abbreviation A.P. is well-understood.
However, use of the term progression, although ubiquitous in the literature, is deprecated on $\mathsf{Pr} \infty \mathsf{fWiki}$ on account of the fact that the term is used both for arithmetic sequence and arithmetic series, and this can be a source of confusion.
Also see
- Results about Arithmetic 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.
Sources
- 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): arithmetic progression
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): arithmetic progression (arithmetic sequence)
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): arithmetic progression (arithmetic sequence)
- 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): arithmetic sequence