Difference between revisions of "Arithmetic sequence"

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An '''arithmetic sequence''' (or '''arithmetic progression''')
An '''arithmetic sequence''' (or '''arithmetic progression''')
is a (finite or infinite) [[sequence]]
is a (finite or infinite) [[sequence]]
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Examples for arithmetic sequences are
Examples for arithmetic sequences are
* 2, 5, 8, 11, 14, 17 (finite, 6 elements, difference 3)
* 2, 5, 8, 11, 14, 17 (finite, length 6: 6 elements, difference 3)
* 5, 1, −3, −7 (finite, 4 elements, difference −4)
* 5, 1, −3, −7 (finite, length 4: 4 elements, difference −4)
* 1, 3, 5, 7, 9, ... (2''i'' − 1), ... (infinite, difference 2)
* 1, 3, 5, 7, 9, ... (2''n'' − 1), ... (infinite, difference 2)


== Mathematical notation ==
== Mathematical notation ==
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: <math> a_1 + a_2 +\cdots+ a_n = \sum_{i=1}^n a_i
: <math> a_1 + a_2 +\cdots+ a_n = \sum_{i=1}^n a_i
         = (a_1 + a_n){n \over 2}
         = (a_1 + a_n){n \over 2}
         = a_1 + d {n(n-1) \over 2}
         = na_1 + d {n(n-1) \over 2}
</math>
</math>

Latest revision as of 12:40, 9 January 2010

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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