1. Definitions Series: an indicated sum of terms of a sequence
Infinite Arithmetic series: goes on forever
Arithmetic series: implies finite series

2. With 100 numbers there are 50 pairs that add up to 101. 50(101) = 5050
Often in applications we will want the sum of a certain number of terms in an arithmetic sequence.
The story is told of a grade school teacher In the 1700's that wanted to keep her class busy while she graded papers so she asked them to add up all of the numbers from 1 to These numbers are an arithmetic sequence with common difference 1. Carl Friedrich Gauss was in the class and had the answer in a minute or two (remember no calculators in those days). This is what he did:
sum is 101
sum is 101
With 100 numbers there are 50 pairs that add up to (101) = 5050

3. Let’s find the sum of 1 + 3 +5 + . . . + 59
This will always work with an arithmetic sequence. The formula for the sum of n terms is:
n is the number of terms so n/2 would be the number of pairs
first term
last term
Let’s find the sum of
But how many terms are there?
We can write a formula for the sequence and then figure out what term number 59 is.

4. 2n - 1 = 59 n = 30 Let’s find the sum of 1 + 3 +5 + . . . + 59
first term
last term
Let’s find the sum of
The common difference is 2 and the first term is one so:
Set this equal to 59 to find n. Remember n is the term number.
2n - 1 = 59 n = 30
So there are 30 terms to sum up.