Sum of the First n Terms of an Arithmetic Series

If a series is arithmetic the sum of the first n terms, denoted Sn, 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,
,
where n is the number of terms, a1 is the first term and an is the last term.

Example 1:

Find the sum of the first 20 terms of the arithmetic series if a1 = 5 and a20 = 62.

Example 2:

Find the sum of the first 40 terms of the arithmetic series 2 + 5 + 8 + 11 + ···.

First find the 40th term:

Then find the sum:

Example 3:

Find the sum:

First find a1 and a50:

  a1 = 3(1) + 2 = 5

a50 = 3(50) + 2 = 152

Then find the sum: