Sum of the First n Terms of an Arithmetic Series

If a series is arithmetic the sum of the first n terms, denoted S n , there are ways to find its sum without actually adding all of the terms.

To find the sum of the first n terms of an arithmetic series use the formula, n terms of an arithmetic sequence use the formula,
S n = n( a 1 + a n ) 2 ,
where n is the number of terms, a 1 is the first term and a n is the last term.

The series 3+6+9+12++30 can be expressed as sigma notation n=1 10 3n . This expression is read as the sum of 3n as n goes from 1 to 10

Example 1:

Find the sum of the first 20 terms of the arithmetic series if a 1 =5 and a 20 =62 .

S 20 = 20(5+62) 2 S 20 =670

Example 2:

Find the sum of the first 40 terms of the arithmetic sequence
2,5,8,11,14,

First find the 40 th term:

a 40 = a 1 +(n1)d =2+39(3)=119

Then find the sum:

S n = n( a 1 + a n ) 2 S 40 = 40(2+119) 2 =2420

Example 3:

Find the sum:

k=1 50 (3k+2)

First find a 1 and a 50 :

a 1 =3(1)+2=5 a 20 =3(50)+2=152

Then find the sum:

S k = k( a 1 + a k ) 2 S 50 = 50(5+152) 2 =3925