The series, ,
is an *arithmetic series*
and the summation

is called the
*arithmetic series summation* .

The summation can be solved as follows:
First, we make the simple variable substitution *i*=*n*-*j*:

Note that the term in the first summation in Equation
is independent of *j*.
Also, the second summation is identical to the left hand side.
Rearranging Equation , and simplifying gives

There is, of course, a simpler way to arrive this answer.
Consider the series, .
and suppose *n* is even.
The sum of the first and last element is *n*+1.
So too is the sum of the second and second-last element,
and the third and third-last element, etc.,
and there are *n*/2 such pairs.
Therefore, .

And if *n* is odd, then ,
where *n*-1 is even.
So we can use the previous result for to get
.

Copyright © 1997 by Bruno R. Preiss, P.Eng. All rights reserved.