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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