Definitions

# arithmetic progression

arithmetic progression: see 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 nth term of the sequence is given by:

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

and in general

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

## Sum (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:

$S_n=a_1+\left(a_1+d\right)+\left(a_1+2d\right)+dotsdots+\left(a_1+\left(n-2\right)d\right)+\left(a_1+\left(n-1\right)d\right)$

$S_n=\left(a_n-\left(n-1\right)d\right)+\left(a_n-\left(n-2\right)d\right)+dotsdots+\left(a_n-2d\right)+\left(a_n-d\right)+a_n.$

Add both sides of the two equations. All terms involving d cancel, and so we're left with:

$2S_n=n\left(a_1+a_n\right).$

Rearranging and remembering that $a_n = a_1 + \left(n-1\right)d$, we get:

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

## 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

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

where $x^\left\{overline\left\{n\right\}\right\}$ 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 $1 times 2 times cdots times n$ is given by the factorial $n!$ and that the product

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

for positive integers $m$ and $n$ is given by

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