Data Structures and Algorithms
with Object-Oriented Design Patterns in Java 
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 © 1998 by Bruno R. Preiss, P.Eng. All rights reserved.
