- Arithmetic progression
mathematics, an arithmetic progression or arithmetic sequence is a sequenceof 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 and the common difference of successive members is "d", then the "n"th term of the sequence is given by::
and in general
um (the arithmetic series)
sumof the components of an arithmetic progression is called an arithmetic series.
Formula (for the arithmetic series)
Express the arithmetic series in two different ways:
Add both sides of the two equations. All terms involving "d" cancel, and so we're left with:
Rearranging and remembering that , we get:
The product of the components of an arithmetic progression with an initial element , common difference , and elements in total, is determined in a closed expression by
where denotes the
rising factorialand denotes the Gamma function. (Note however that the formula is not valid when is a negative integer or zero).
This is a generalization from the fact that the product of the progression is given by the
factorialand that the product
positive integers and is given by
Generalized arithmetic progression
Infinite arithmetic series
Thomas Robert Malthus
Primes in arithmetic progression
Problems involving arithmetic progressions
Kahun Papyrus, Rhind Mathematical Papyrus
Ergodic Ramsey theory
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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