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

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