# Arithmetic progression

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Arithmetic progression

In mathematics, an arithmetic progression or arithmetic sequence is a sequence of numbers such that the difference of any two successive members of the sequence is a constant. For instance, the sequence 3, 5, 7, 9, 11, 13... is an arithmetic progression with common difference 2.

If the initial term of an arithmetic progression is $a_1$ and the common difference of successive members is "d", then the "n"th term of the sequence is given by::$a_n = a_1 + \left(n - 1\right)d,$

and in general

:$a_n = a_m + \left(n - m\right)d.$

um (the arithmetic series)

The sum of the components of an arithmetic progression is called an arithmetic series.

Formula (for the arithmetic series)

Express the arithmetic series in two different ways:

$S_n=a_1+\left(a_1+d\right)+\left(a_1+2d\right)+dotsdots+\left(a_1+\left(n-2\right)d\right)+\left(a_1+\left(n-1\right)d\right)$

$S_n=\left(a_n-\left(n-1\right)d\right)+\left(a_n-\left(n-2\right)d\right)+dotsdots+\left(a_n-2d\right)+\left(a_n-d\right)+a_n.$

Add both sides of the two equations. All terms involving "d" cancel, and so we're left with:

$2S_n=n\left(a_1+a_n\right).$

Rearranging and remembering that $a_n = a_1 + \left(n-1\right)d$, we get:

$S_n=frac\left\{n\left( a_1 + a_n\right)\right\}\left\{2\right\}=frac\left\{n \left[ 2a_1 + \left(n-1\right)d\right] \right\}\left\{2\right\}.$

Product

The product of the components of an arithmetic progression with an initial element $a_1$, common difference $d$, and $n$ elements in total, is determined in a closed expression by

:$a_1a_2cdots a_n = d^n \left\{left\left(frac\left\{a_1\right\}\left\{d\right\} ight\right)\right\}^\left\{overline\left\{n = d^n frac\left\{Gamma left\left(a_1/d + n ight\right) \right\}\left\{Gamma left\left( a_1 / d ight\right) \right\},$

where $x^\left\{overline\left\{n$ denotes the rising factorial and $Gamma$ denotes the Gamma function. (Note however that the formula is not valid when $a_1/d$ is a negative integer or zero).

This is a generalization from the fact that the product of the progression $1 imes 2 imes cdots imes n$ is given by the factorial $n!$ and that the product

:$m imes \left(m+1\right) imes \left(m+2\right) imes cdots imes \left(n-2\right) imes \left(n-1\right) imes n ,!$

for positive integers $m$ and $n$ is given by

:$frac\left\{n!\right\}\left\{\left(m-1\right)!\right\}.$

ee also

* Geometric progression
* Generalized arithmetic progression
* Green–Tao theorem
* Infinite arithmetic series
* Thomas Robert Malthus
* Primes in arithmetic progression
* Problems involving arithmetic progressions
* Kahun Papyrus, Rhind Mathematical Papyrus
* Ergodic Ramsey theory

References

*cite book
title = Fibonacci's Liber Abaci
author = Sigler, Laurence E. (trans.)
publisher = Springer-Verlag
year = 2002
id = ISBN 0-387-95419-8
pages = 259–260