, an arithmetic progression
or arithmetic sequence
is a sequence
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
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:
Add both sides of the two equations. All terms involving d cancel, and so we're left with:
Rearranging and remembering that , we get:
The product of the components of an 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. (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 and that the product
for positive integers and is given by