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:
If S n represents the sum of an arithmetic sequence with terms , then
This formula requires the values of the first and last terms and the number of terms.
Substituting this last expression for ( a 1 + a n ) into Formula 1, another formula for the sum of an arithmetic sequence is formed.
This formula for the sum of an arithmetic sequence requires the first term, the common difference, and the number of terms.
In the arithmetic sequence –3, 4, 11, 18, …, find the sum of the first 20 terms.
Use Formula 2 to find the sum.
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.
Now, use Formula 1.
The sum of the multiples of 3 between 28 and 112 is 1974.