Arithmetic sequence

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[edit intro]

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

## Mathematical notation[edit]

A finite sequence

a1,a2,…,an={ai∣i=1,…,n}={ai}i=1,…,n

or an infinite sequence

a0,a1,a2,…={ai∣i∈ℕ}={ai}i∈ℕ

is called arithmetic sequence if

ai+1−ai=d

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

### General form[edit]

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

ai=a1+(i−1)d

### Sum[edit]

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

a1+a2+⋯+an=∑i=1nai=(a1+an)n2=na1+dn(n−1)2