Arithmetic sequence

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, ... (2i − 1), ... (infinite, difference 2)

Mathematical notation

A finite sequence

$a_{1},a_{2},\dots ,a_{n}=\{a_{i}\mid i=1,\dots ,n\}=\{a_{i}\}_{i=1,\dots ,n}$

or an infinite sequence

$a_{0},a_{1},a_{2},\dots =\{a_{i}\mid i\in \mathbb {N} \}=\{a_{i}\}_{i\in \mathbb {N} }$

is called arithmetic sequence if

$a_{i+1}-a_{i}=d$

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

$a_{i}=a_{1}+(i-1)d$

Sum

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

$a_{1}+a_{2}+\cdots +a_{n}=\sum _{i=1}^{n}a_{i}=(a_{1}+a_{n}){n \over 2}=a_{1}+d{n(n-1) \over 2}$