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

and in general

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

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:

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

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

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:

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

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

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

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 $1\; imes\; 2\; imes\; cdots\; imes\; n$ is given by the factorial $n!$ and that the product

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

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

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

ee also

* 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

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

External links

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