There are many articles and books on the calculus of finite differences. One nice one is: Martin Gardner, The calculus of finite differences, in Martin Gardner's New Mathematical Diversions from Scientific American, Simon and Schuster, New York, 1971, chapter 20.
If you have a sequence and you are trying to figure out what the rule
is, or if the sequence is already known elsewhere, the unbeatable
place to turn to is the website of Neil Sloane,
Sloane's On-Line Encyclopedia of Integer Sequences
(http://www.research.att.com/~njas/sequences/).
Many summation formulas can be found with Maple. For example, type
sum(i^5,i=1..n);
and the result is
Or you get the result in factored form by typing
factor(sum(i^5,i=1..n));
which yields
