Here is a formula for the sum of the first n natural numbers:
One example of this is 
. Gauss as a small
child is said to have calculated this by noticing that 1+100=101,
2+99=101, 3+98=101, etc., so the entire sum is 50(101)=5050.
We can use this idea to prove the formula. Let S be the sum. Then
Adding these together yields
So S=n(n+1)/2.