arithmetic progression: Definition from Answers.com

In mathematics, an arithmetic progression (A.P.) 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 nth term of the sequence is given by:

$\ a_n = a_1 + (n - 1)d,$

and in general

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

A finite portion of an arithmetic progression is called a finite arithmetic progression and sometimes just called an arithmetic progression.

1 Sum (the arithmetic series)
2 Product
3 See also
4 References
5 External links

Sum (the arithmetic series)

The sum of the members of a finite arithmetic progression is called an arithmetic series.

Express the arithmetic series in two different ways:

$S_n=a_1+(a_1+d)+(a_1+2d)+\cdots+(a_1+(n-2)d)+(a_1+(n-1)d)$

$S_n=(a_n-(n-1)d)+(a_n-(n-2)d)+\cdots+(a_n-2d)+(a_n-d)+a_n.$

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:

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

This formula has long been known, but Carl Friedrich Gauss is said to have rediscovered it at the age of eight. However, there are reasons for skepticism concerning this story.^[1]

Product

The product of the members of a finite arithmetic progression with an initial element $a 1$ , common difference $d$ , and $n$ elements in total, is determined in a closed expression by

$a_1a_2\cdots a_n = d^n {\left(\frac{a_1}{d}\right)}^{\overline{n}} = d^n \frac{\Gamma \left(a_1/d + n\right) }{\Gamma \left( a_1 / d \right) },$

where $x^{\overline{n}}$ denotes the rising factorial and $Γ$ 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

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

References

^ Hayes, Brian, "Gauss's Day of Reckoning", American Scientist, http://www.americanscientist.org/issues/id.3483,y.0,no.,content.true,page.1,css.print/issue.aspx .

Sigler, Laurence E. (trans.) (2002). Fibonacci's Liber Abaci. Springer-Verlag. pp. 259–260. ISBN 0-387-95419-8.

External links

Derivation of formula for sum of arithmetic progression at Mathalino.com
Weisstein, Eric W., "Arithmetic progression" from MathWorld.
Weisstein, Eric W., "Arithmetic series" from MathWorld.

This entry is from Wikipedia, the leading user-contributed encyclopedia. It may not have been reviewed by professional editors (see full disclaimer)

Donate to Wikimedia

	arithmetic series (mathematics)
	common difference (mathematics)
	harmonic progression (mathematics)

	What is the uses of arithmetic progression? Read answer...
	Where can Arithmetic Progressions be used? Read answer...
	Practical use of arithmetic progression? Read answer...

	Who was the founder of arithmetic progression?
	Designs using arithmetic progression?
	History of arithmetic progression?

		Dictionary. The American Heritage® Dictionary of the English Language, Fourth Edition Copyright © 2007, 2000 by Houghton Mifflin Company. Updated in 2009. Published by Houghton Mifflin Company. All rights reserved. Read more
		WordNet. WordNet 1.7.1 Copyright © 2001 by Princeton University. All rights reserved. Read more
		Wikipedia. This article is licensed under the Creative Commons Attribution/Share-Alike License. It uses material from the Wikipedia article "Arithmetic progression". Read more

arithmetic progression

Contents

Sum (the arithmetic series)

Product

See also

References

External links