Arithmetic progression

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Arithmetic progression

In mathematics

Mathematics

Mathematics is the study of quantity, structure, space, change, and related topics of pattern and form. Mathematicians seek out patterns whether found in numbers, space, natural science, computers, imaginary abstractions, or elsewhere....
, an arithmetic progression (A.P.) or arithmetic sequence is a sequence

Sequence

In mathematics, a sequence is an ordered list of objects . Like a Set , it contains Element , and the number of terms is called the length of the sequence....
of number

Number

A number is a mathematical object used in counting and measurement. A notational symbol which represents a number is called a Numeral system, but in common usage the word number is used for both the abstract object and the symbol, as well as for the numeral for the number....
s 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 nth term of the sequence is given by:

and in general

A finite portion of an arithmetic progression is called a finite arithmetic progression and sometimes just called an arithmetic progression.

Sum (the arithmetic series)
The sum

SUM

SUM can refer to:* The State University of Management* Soccer United Marketing* StartUp-Manager...
of the members of a finite arithmetic progression is called an arithmetic series.

Express the arithmetic series in two different ways

Double counting (proof technique)

In combinatorics, double counting, also called counting in two ways, is a combinatorial proof technique that involves counting the size of a Set in two ways in order to show that the two resulting expressions for the size of the set are equal....
:

Add both sides of the two equations.

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Sum (the arithmetic series)

The sum

SUM

Double counting (proof technique)

Product

The product

Product (mathematics)

In the a mathematics, a product is the result of Multiplication, or an expression that identifies divisors to be multiplied. The order in real number or complex number numbers are multiplied has no bearing on the product; this is known as the Commutativity of multiplication....
of the members of a finite arithmetic progression with an initial element , common difference , and elements in total, is determined in a closed expression by

where denotes the rising factorial and denotes the Gamma function

Gamma function

In mathematics, the Gamma function is an extension of the factorial function to real number and complex number numbers. For a complex number z with positive real part the Gamma function is defined by...
. (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 factorial

Factorial

In mathematics, the factorial of a negative and non-negative numbers integer n, denoted by n!, is the Product of all positive integers less than or equal to n....
and that the product

for positive integers and is given by

Arithmetic progression

Sum (the arithmetic series)

Product

See also

External links