If we have to add up all the terms of a sequence, for example
it is convenient to be able to express this in a compact form. We can write the sum of the above terms as
In English this means, “add up the k+1 powers of 2 starting with the zeroth power.”
The summation has several convenient properties. If A and B are constants and a-n and b-n are sequences then
(1)
(2)
These may be used in the following way:
If
and
find in terms of
Using (2) and (1) in turn we obtain
The process outlined above can be used to find the sum of terms of any polynomial sequence by expressing the function as a sum of powers of n. The sum of each power can be looked up in a table hence the sum of any polynomial found, either as a number or a function of n.