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arithmetic progression: see progression.

The Columbia Electronic Encyclopedia Copyright © 2004.

Licensed from Columbia University Press

Licensed from Columbia University Press

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.## Sum (the arithmetic series)

#### Formula (for the arithmetic series)

Express the arithmetic series in two different ways:## Product

## See also

## References

## External links

If the initial term of an arithmetic progression is $a\_1$ and the common difference of successive members is d, then the nth term of the sequence is given by:

- $a\_n\; =\; a\_1\; +\; (n\; -\; 1)d,$

and in general

- $a\_n\; =\; a\_m\; +\; (n\; -\; m)d.$

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

$S\_n=a\_1+(a\_1+d)+(a\_1+2d)+dotsdots+(a\_1+(n-2)d)+(a\_1+(n-1)d)$

$S\_n=(a\_n-(n-1)d)+(a\_n-(n-2)d)+dotsdots+(a\_n-2d)+(a\_n-d)+a\_n.$

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

$2S\_n=n(a\_1+a\_n).$

Rearranging and remembering that $a\_n\; =\; a\_1\; +\; (n-1)d$, we get:

$S\_n=frac\{n(a\_1\; +\; a\_n)\}\{2\}=frac\{n[2a\_1\; +\; (n-1)d]\}\{2\}.$

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

- $a\_1a\_2cdots\; a\_n\; =\; d^n\; \{left(frac\{a\_1\}\{d\}right)\}^\{overline\{n\}\}\; =\; d^n\; frac\{Gamma\; left(a\_1/d\; +\; nright)\; \}\{Gamma\; left(a\_1\; /\; d\; right)\; \},$

where $x^\{overline\{n\}\}$ 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 $1\; times\; 2\; times\; cdots\; times\; n$ is given by the factorial $n!$ and that the product

- $m\; times\; (m+1)\; times\; (m+2)\; times\; cdots\; times\; (n-2)\; times\; (n-1)\; times\; n\; ,!$

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

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

- Addition
- 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

- Sigler, Laurence E. (trans.) (2002).
*Fibonacci's Liber Abaci*. Springer-Verlag. ISBN 0-387-95419-8.

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Last updated on Sunday September 28, 2008 at 14:48:16 PDT (GMT -0700)

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