Arithmetic Series

An arithmetic series is the sum of the terms in an arithmetic sequence with a definite number of terms. Following is a simple formula for finding the sum:

Formula 1:

If S n represents the sum of an arithmetic sequence with terms equation, then equation

This formula requires the values of the first and last terms and the number of terms.

equation

Substituting this last expression for ( a 1 + a n ) into Formula 1, another formula for the sum of an arithmetic sequence is formed.

Formula 2: equation

This formula for the sum of an arithmetic sequence requires the first term, the common difference, and the number of terms.

Example 1

In the arithmetic sequence –3, 4, 11, 18, …, find the sum of the first 20 terms.

equation

Use Formula 2 to find the sum.

equation

Example 2

Find the sum of the multiples of 3 between 28 and 112.

The first multiple of 3 between 28 and 112 is 30, and the last multiple of 3 between 28 and 112 is 111. In order to use Formula 1, the number of terms must be known. a n = a 1 + ( n – 1) d can be used to find n.

equation

Now, use Formula 1.

equation

The sum of the multiples of 3 between 28 and 112 is 1974.