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 + (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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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


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