This is why the sum of n integers is 1/2n(n+1). Someone called Gauss worked it out in his head at school, but I like this picture.
If you make a rectangle n wide and n+1 high, and shade half of it across the diagonal, you can see that the light squares are the integers up to n (1, 2, 3…n).
The area of the rectangle is n*(n+1), and half of that is the row of integers.
This proof is mentioned in The Mathematical Universe (William Dunham)
