# Arithmetic progression

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

*
*

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### Look at other dictionaries:

• arithmetic progression — n. a sequence of terms each of which, after the first, is derived by adding to the preceding one a common difference [5, 9, 13, 17, etc. form an arithmetic progression] …   English World dictionary

• arithmetic progression — (also arithmetic series) ► NOUN ▪ a sequence of numbers in which each differs from the preceding one by a constant quantity (e.g. 1, 2, 3, 4, etc.; 9, 7, 5, 3, etc.) …   English terms dictionary

• arithmetic progression — noun (mathematics) a progression in which a constant is added to each term in order to obtain the next term 1 4 7 10 13 is the start of an arithmetic progression • Topics: ↑mathematics, ↑math, ↑maths • Hypernyms: ↑progression, ↑ …   Useful english dictionary

• arithmetic progression — noun A sequence in which each term except the first is obtained from the previous by adding a constant value, known as the common difference of the arithmetic progression …   Wiktionary

• arithmetic progression — arithmetic pro gression n a set of numbers in order of value in which a particular number is added to each to produce the next (as in 2, 4, 6, 8, ...) →↑geometric progression …   Dictionary of contemporary English

• arithmetic progression — a sequence in which each term is obtained by the addition of a constant number to the preceding term, as 1, 4, 7, 10, 13, and 6, 1, 4, 9, 14. Also called arithmetic series. [1585 95] * * * …   Universalium

• arithmetic progression — (also arithmetic series) noun a sequence of numbers in which each differs from the preceding one by a constant quantity (e.g. 1, 2, 3, 4, etc.; 9, 7, 5, 3, etc.) …   English new terms dictionary

• arithmetic progression — noun Date: 1594 a progression (as 3, 5, 7, 9) in which the difference between any term and its predecessor is constant …   New Collegiate Dictionary

• arithmetic progression — arith,metic pro gression noun singular a series of numbers in which the same number is added to each number to produce the next, for example 3, 6, 9, 12 ─ compare GEOMETRIC PROGRESSION …   Usage of the words and phrases in modern English

• arithmetic progression — noun (C) a set of numbers in order of value in which a particular number is added to each to produce the next (as in 2, 4, 6, 8 ...) compare geometric progression …   Longman dictionary of contemporary English