The Gauss Summation offers a quick way to add up the first n counting numbers:
1 + 2 + 3 + ... + n
Picture this as a staircase made of square blocks. The first column is 1 block tall, the second is 2 blocks tall, and so on, ending with a column n blocks tall.
Now, imagine drawing the biggest possible triangle that fits inside this staircase. Its area comes out to nยฒ/2. What's left over are n tiny half-squares, each with an area of 1/2, giving a total leftover area of n/2.
Add the triangle and the leftovers together, and you get the total number of blocks in the staircaseโwhich is exactly the sum 1 + 2 + ... + n.