According to an old story, one day Gauss and his classmates were asked to find the sum of these first hundred counting numbers. All the other students in the class began by adding two numbers at a time, starting from the first term. This is a natural reaction. Also, it is a valid approach, although it is not the most efficient method, as we will see. But Gauss found a quicker way. First, he wrote the sum twice. By adding vertically, each pair of numbers adds up to 101.

Since there are 100 of these sums of 101, the total is 100101 = 10,100. Because this sum 10,100 is twice the sum of the numbers 1 through 100, we have: