An old story says Gauss found the sum of the above problem, and at the same time developed a formula, in less than a few minutes when he was a young student.

The objective of this section is to find a formula for calculating the sum of the first n terms of any arithmetic sequence. As will be shown, the method we are using today was developed by the mathematician Carl F. Gauss (1777 – 1855) when he was a young student.

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 will be seen. But Gauss found a quicker way. First, he wrote the sum twice, one in an ordinary order and the other in a reverse order:

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:

1 + 2 + 3 + . . . + 98 + 99 + 100 = 100101 / 2 = 5050.

Using Gauss´ approach, we can find the formula for the sum of the first n terms of an arithmetic sequence having first term a₁ and a common difference c, as follows:

in which it is understood that k is to be replaced by each of the numbers 1, 2, 3, 4, and 5 in the expression 2k, and then the sum of the resulting values is to be indicated.

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