Finding the Sum of the Arithmetic Sequence

An arithmetic sequence is a sequence of numbers such that the difference between two consecutive terms is constant.  The sequence

7, 13, 19, 25, 31, 37, 43, 59

is an example of an arithmetic sequence with first term 7, constant difference 6, and last term 49.

You have learned in that the formula for finding the nth term of the arithmetic sequence a_n with first term a_1, and constant difference d is given by

a_n = a_1 + (n-1)d .

In this post, we derive the formula for finding the sum of all the numbers in an arithmetic sequence. We take the specific example above and use Gauss’ method in finding the sum of the first 100 positive integers. Recall that in adding the first 100 integers, Gauss added the first integer to the last, the second integer to  the second to the last, the third integer and the third to the last and so on. » Read more