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

${\displaystyle a_1,a_2,\dots ,a_n=\{a_i\mid i=1,\dots ,n\}=\{a_i{\}}_{i=1,\dots ,n}}$

or an infinite sequence

${\displaystyle a_0,a_1,a_2,\dots =\{a_i\mid i\in \mathbb{\mathbb{N}}\}=\{a_i{\}}_{i\in \mathbb{\mathbb{N}}}}$

is called arithmetic sequence if

${\displaystyle a_{i+1}-a_i=d}$

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

${\displaystyle a_i=a_1+(i-1)d}$

Sum[edit]

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

${\displaystyle a_1+a_2+\cdots +a_n={\displaystyle \sum _{i=1}^n}{a}_i=(a_1+a_n)\frac{n}{2}=n{a}_1+d\frac{n(n-1)}{2}}$