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

A finite sequence

or an infinite sequence

is called arithmetic sequence if

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

Sum

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