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

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 100. 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 1 + 2 + 3 + 4 + 5 + . . . + 96 + 97 + 98 + 99 + 100 sum is 101 With 100 numbers there are 50 pairs that add up to 101. 50(101) = 5050

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 1 + 3 +5 + . . . + 59 But how many terms are there? We can write a formula for the sequence and then figure out what term number 59 is.

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 1 + 3 +5 + . . . + 59 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.