Personal Blog By Nilesh: Arithmetic Series

Sum (the arithmetic series)

The sum of the components of an arithmetic progression is called an arithmetic series.

[edit] Formula (for the arithmetic series)

Express the arithmetic series in two different ways:

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

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

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 an = a1 + (n − 1)d, we get:

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

[edit] Product

The product of the components of an arithmetic progression with an initial element a1, 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 a1 / 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