Arithmetic sequence

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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

a 1 , a 2 , … , a n = { a i ∣ i = 1 , … , n } = { a i } i = 1 , … , n {\displaystyle 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 , ⋯ = { a i ∣ i ∈ N } = { a i } i ∈ N {\displaystyle 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 {\displaystyle a_{i+1}-a_{i}=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

a i = a 1 + ( i − 1 ) d {\displaystyle a_{i}=a_{1}+(i-1)d}

### Sum[edit]

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

a 1 + a 2 + ⋯ + a n = ∑ i = 1 n a i = ( a 1 + a n ) n 2 = n a 1 + d n ( n − 1 ) 2 {\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}}