Question

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.

Answer 1

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**