# Arithmetic sequence

From Citizendium, the Citizens' Compendium

An **arithmetic sequence** (or **arithmetic progression**)
is a (finite or infinite) sequence
of (real or complex) numbers
such that the difference of consecutive elements is the same for each pair.

Examples for arithmetic sequences are

- 2, 5, 8, 11, 14, 17 (finite, length 6: 6 elements, difference 3)
- 5, 1, −3, −7 (finite, length 4: 4 elements, difference −4)
- 1, 3, 5, 7, 9, ... (2
*n*− 1), ... (infinite, difference 2)

## Mathematical notation

A finite sequence

or an infinite sequence

is called arithmetic sequence if

for all indices *i*. (The index set need not start with 0 or 1.)

### General form

Thus, the elements of an arithmetic sequence can be written as

### Sum

The sum (of the elements) of a finite arithmetic sequence is