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

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

ai + 1ai = 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

ai = a1 + (i − 1)d


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} = na_1 + d {n(n-1) \over 2}
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