Copyright © 2014, 2010, 2007 Pearson Education, Inc. Chapter 8 Sequences, Induction, and Probability 8.2 Arithmetic Sequences Copyright © 2014, 2010, 2007 Pearson Education, Inc. 1

Objectives: Find the common difference for an arithmetic sequence. Write terms of an arithmetic sequence. Use the formula for the general terms of an arithmetic sequence. Use the formula for the sum of the first n terms of an arithmetic sequence.

Definition of an Arithmetic Sequence An arithmetic sequence is a sequence in which each term after the first differs from the preceding term by a constant amount. The difference between consecutive terms is called the common difference of the sequence.

Example: Writing the Terms of an Arithmetic Sequence Write the first six terms of the arithmetic sequence in which a1 = 100 and an = an–1 – 30. The terms are 100, 70, 40, 10, –20, –50.

General Term of an Arithmetic Sequence The nth term (the general term) of an arithmetic sequence with first term a1 and common difference d is

Example: Using the Formula for the General Term of an Arithmetic Sequence Find the ninth term of the arithmetic sequence whose first term is 6 and whose common difference is –5. To find the ninth term, a9, we replace n in the formula with 9, a1 with 6, and d with –5. The ninth term is –34.

Example: Using an Arithmetic Sequence to Model Changes in the U. S Example: Using an Arithmetic Sequence to Model Changes in the U.S. Population The data in the graph show that in 2010, 16% of the U.S. population was Latino. On average, this is projected to increase by approximately 0.35% per year. Write a formula for the nth term of the arithmetic sequence that describes the percentage of the U.S. population that will be Latino n years after 2009.

Example: Using an Arithmetic Sequence to Model Changes in the U. S Example: Using an Arithmetic Sequence to Model Changes in the U.S. Population (continued) The formula for the nth term of the arithmetic sequence that describes the percentage of the U.S. population that will be Latino n years after 2009 is

We have found that the formula for the nth term of the Example: Using an Arithmetic Sequence to Model Changes in the U.S. Population (continued) We have found that the formula for the nth term of the arithmetic sequence that describes the percentage of the U.S. population that will be Latino n years after 2009 is What percentage of the U.S. population is projected to be Latino in 2030? 23% of the U.S. population is projected to be Latino in 2030.

The Sum of the First n Terms of an Arithmetic Sequence The sum, Sn ,of the first n terms of an arithmetic sequence is given by in which a1 is the first term and an is the nth term.

Example: Finding the Sum of n Terms of an Arithmetic Sequence Find the sum of the first 15 terms of the arithmetic sequence: 3, 6, 9, 12, … We use the formula for the general term of a sequence to find a15. The common difference is 3. The sum of the first 15 terms of the sequence is 360.