From Citizendium - Reading time: 1 min
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
![{\displaystyle a_{1},a_{2},\dots ,a_{n}=\{a_{i}\mid i=1,\dots ,n\}=\{a_{i}\}_{i=1,\dots ,n}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7af989115ff1a575e297db55052e733a0c8f8f5f)
or an infinite sequence
![{\displaystyle a_{0},a_{1},a_{2},\dots =\{a_{i}\mid i\in \mathbb {N} \}=\{a_{i}\}_{i\in \mathbb {N} }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/feaf9305fb52de33fc505c537d26cfd641e7fc4e)
is called arithmetic sequence if
![{\displaystyle a_{i+1}-a_{i}=d}](https://wikimedia.org/api/rest_v1/media/math/render/svg/14c4744c6bf844c15bb086587eca7a6dae245313)
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
![{\displaystyle a_{i}=a_{1}+(i-1)d}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b9a3587e67e8cab800bbc4c21557d4d1d1bcb2e9)
The sum (of the elements) of a finite arithmetic sequence is
![{\displaystyle a_{1}+a_{2}+\cdots +a_{n}=\sum _{i=1}^{n}a_{i}=(a_{1}+a_{n}){n \over 2}=na_{1}+d{n(n-1) \over 2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3e417787103e5034a899739a9cff125c6185ad8e)