How does:
(1 + 2 + ... + N) / N = (N + 1) / 2
or
(1 + 2 + ... + N + N) / N = (N + 3) / 2
My textbook says this is elementary math but I have forgotten the method for finding the answer.
The example you gave is called an arithmetic sequence, not a geometric sequence.
A simple way to convince yourself that the result is correct is to write the same sequence backwards, add it to itself, and divide by 2:
1 + 2 + 3 + ... + N-1 + N = S
+ N + N-1 + N-2 + ... + 2 + 1 = S
--------------------------------------
N+1 + N+1 + N+1 + ... + N+1 + N+1 = 2S
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
N terms
= (N+1)*N = 2S
(N+1)*N/2 = 2S/2 = S =
**S = (N+1)*N/2**