Arithmetic progression

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In mathematics, an arithmetic progression (A.P.) 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 nth term of the sequence is given by:

$\ a_n = a_1 + (n - 1)d,$

and in general

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

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

[edit] Sum (the arithmetic series)

The sum of the members of a finite arithmetic progression is called an arithmetic series.

Express the arithmetic series in two different ways:

$S_n=a_1+(a_1+d)+(a_1+2d)+\cdots+(a_1+(n-2)d)+(a_1+(n-1)d)$

$S_n=(a_n-(n-1)d)+(a_n-(n-2)d)+\cdots+(a_n-2d)+(a_n-d)+a_n.$

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:

$S_n=\frac{n( a_1 + a_n)}{2}=\frac{n[ 2a_1 + (n-1)d]}{2}.$

This formula has long been known, but Carl Friedrich Gauss is said to have rediscovered it at the age of eight. However, there are reasons for skepticism concerning this story.^[1]

[edit] Product

The product of the members of a finite arithmetic progression with an initial element $a 1$ , common difference $d$ , and $n$ elements in total, is determined in a closed expression by

$a_1a_2\cdots a_n = d^n {\left(\frac{a_1}{d}\right)}^{\overline{n}} = d^n \frac{\Gamma \left(a_1/d + n\right) }{\Gamma \left( a_1 / d \right) },$

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).

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