NationMaster - Encyclopedia: 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. For other meanings of mathematics or uses of math and maths, see Mathematics (disambiguation) and Math (disambiguation). ... For other senses of this word, see sequence (disambiguation). ... For other uses, see Number (disambiguation). ...

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

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

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

Sum (the arithmetic series)

The sum of the components of an arithmetic progression is called an arithmetic series. Addition is one of the basic operations of arithmetic. ...

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

$a_1a_2cdots a_n = d^n {left(frac{a_1}{d}right)}^{overline{n}} = d^n frac{Gamma left(a_1/d + nright) }{Gamma left( a_1 / d right) },$

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). In mathematics, the Pochhammer symbol is used in the theory of special functions to represent the rising factorial or upper factorial and, confusingly, is used in combinatorics to represent the falling factorial or lower factorial The empty product (x)0 is defined to be 1 in both cases. ... The Gamma function along part of the real axis In mathematics, the Gamma function (represented by the capitalized Greek letter Î“) is an extension of the factorial function to real and complex numbers. ...

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 For factorial rings in mathematics, see unique factorisation domain. ...

$m times (m+1) times (m+2) times cdots times (n-2) times (n-1) times n ,!$

for positive integers $m$ and $n$ is given by Natural number can mean either a positive integer (1, 2, 3, 4, ...) or a non-negative integer (0, 1, 2, 3, 4, ...). Natural numbers have two main purposes: they can be used for counting (there are 3 apples on the table), or they can be used for ordering (this is...

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

References

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

External links

Eric W. Weisstein, Arithmetic progression at MathWorld.
Eric W. Weisstein, Arithmetic series at MathWorld.

Categories: Mathematical series | Sequences and series | Articles containing proofs

Eric W. Weisstein (born March 18, 1969, in Bloomington, Indiana) is an encyclopedist who created and maintains MathWorld and Eric Weissteins World of Science (ScienceWorld). ... MathWorld is an online mathematics reference work, sponsored by Wolfram Research Inc. ... Eric W. Weisstein (born March 18, 1969, in Bloomington, Indiana) is an encyclopedist who created and maintains MathWorld and Eric Weissteins World of Science (ScienceWorld). ... MathWorld is an online mathematics reference work, sponsored by Wolfram Research Inc. ...

Results from FactBites:

Arithmetic - Wikipedia, the free encyclopedia (341 words)
	Arithmetic or arithmetics (from the Greek word ἀριθμός = number) in common usage is a branch of (or the forerunner of) mathematics which records elementary properties of certain operations on numerals, though professional mathematicians often treat arithmetic as a synonym for number theory.
	Primary education in mathematics often places a strong focus on arithmetic, as further studies in mathematics as well as science benefit from an understanding of arithmetic.
	The arithmetic of natural numbers, integers, rational numbers (in the form of vulgar fractions), and real numbers (using the decimal place-value system known as algorism) is typically studied by schoolchildren, who learn manual algorithms for arithmetic.

arithmetic progression - definition of arithmetic progression in Encyclopedia (244 words)
	In mathematics, an arithmetic progression is a sequence of numbers such that the difference of any two successive members of the sequence is a constant.
	If the initial term of an arithmetic progression is a and the common difference of successive members is d, then the n-th term of the sequence is given by
	The sum of the numbers in (an initial segment of) an arithmetic progression is sometimes called an arithmetic series.

More results at FactBites »

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Contents

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Formula (for the arithmetic series)

Product

See also

References

External links

Sum (the arithmetic series)