I think you are just messing up a lot of the coefficients and that is why you are getting different answers! Here is how to do it properly. The geometric sum is:
In your case you have , so your sequence will sum to
Now, your term is but you have changed the power (you have added 1). Hence you are now summing the following
In the first case you have , and you get
In the second case you have , and you get (after a slight simplification)
which lo and behold is the same result. The bottom line is that what you have done is correct in the sense that there is more than one way to do this problem. However you are incorrect in your use of the formulas.
The best formula to use is:
Then if you change the power on the r, you need to change the indices of summation. To help you do that, here is an example. It doesn't have the a, but the a is not important, since it is a constant and if you need it you can multiply the result by it!