TITLE: Where do summation formulas come from?
QUESTION [16 upvotes]: It's a classic problem in an introductory proof course to prove that $\sum_{ i \mathop =1}^ni = \frac{n(n+1)}{2}$ by induction. The problem with induction is that you can't prove what the sum is unless you already have an idea of what it should be. I would like to know what the process is for getting the idea. 
Wikipedia has plenty of summation formulas listed, and there are surely lots more, but I think I should be able to simplify summations without referring to a table. I don't suppose there's a universal technique for deriving all of them, but it would be good to know at least a few things to try.
This question was motivated by an answer involving summation, and while I have no doubt that it's true, I wouldn't know how to get the answer to the particular summation without being told beforehand.

REPLY [2 votes]: May be with an example it might be more clear. Take for example 
$$1+2+...+50$$
As you can remark, you have that 
$$51=1+50=2+49=3+48=4+47=...$$
and thus, in $$1+...+n,$$
if $n$ is even, you sum $$\underbrace{(n+1)+...+(n+1)}_{\frac{n}{2}\ times}=\frac{n(n+1)}{2}.$$
If $n$ is odd, you sum
$$\underbrace{(n+1)+...+(n+1)}_{\frac{(n-1)}{2}\ times}+\frac{n+1}{2}=\frac{(n+1)(n-1)+n+1}{2}=\frac{(n+1)(n-1+1)}{2}=\frac{n(n+1)}{2}$$