The sum S can also be expressed as

As a first step, pull out a factor of 5 from the sum:

Shift the index to force the sum to start at k = 0, then distribute the summation:

The second sum is geometric, with

This leaves us with

For the remaining sum, add and subtract the k = n-th term, so that we have

Then in the sum, we get 0 for the k = 0 term and end up recovering another copy of S :

Solving for S gives
