How do I solve an arithmetic sequence?

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开发者_如何学Go 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**

Mathematical induction. http://en.wikipedia.org/wiki/Mathematical_induction#Example

The second claim you stated follows from the first by just adding N / N = 1 = 2 / 2.

Sum of n natural numbers is denoted by n(n+1)/2.

So the given the first problem you have correctly mentioned the output will (n+1)/2.

for the second problem.

the solution is (n(n+1)/2n)+n/n = (n+1)/2 +1 = (n+3)/2. You would observer actual series is sum of n natural numbers plus n. So thats how i split the terms.