1. Slide 11.2- 1 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

2. OBJECTIVES Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Arithmetic Sequences; Partial Sums Learn to identify an arithmetic sequence and find its common difference. Learn to sum the first n terms of an arithmetic sequence. SECTION 11.2 1 2

3. Slide 11.2- 3 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley DEFINITION OF AN ARITHMETIC SEQUENCE The sequence a 1, a 2, a 3, a 4, …, a n, … is an arithmetic sequence, or an arithmetic progression if there is a number d such that each term in the sequence except the first is obtained from the preceding term by adding d to it. The number d is called the common difference of the arithmetic sequence. We have d = a n + 1 – a n, n ≥ 1.

4. Slide 11.2- 4 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley RECURSIVE DEFINITION OF AN ARITHMETIC SEQUENCE An arithmetic sequence a 1, a 2, a 3, a 4, …, a n, … can be defined recursively. The recursive formula a n + 1 = a n + d for n ≥ 1 defines an arithmetic sequence with first term a 1 and common difference d.

5. Slide 11.2- 5 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley n TH TERM OF AN ARITHMETIC SEQUENCE If a sequence a 1, a 2, a 3, … is an arithmetic sequence, then its nth term, a n, is given by a n = a 1 + (n – 1)d, where a 1 is the first term and d is the common difference.

6. Slide 11.2- 6 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 3 Finding the Common Difference of an Arithmetic Sequence Find the common difference d and the nth term a n of an arithmetic sequence whose 5th term is 15 and whose 20th term is 45. Solution

7. Slide 11.2- 7 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 3 Finding the Common Difference of an Arithmetic Sequence Solution continued The nth term is given by a n = 2n + 5, n ≥ 1. gives a 1 = 7 and d = 2. Solving the system of equations

8. Slide 11.2- 8 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley SUM OF n TERMS OF AN ARITHMETIC SEQUENCE Let a 1, a 2, a 3, … a n be the first n terms of an arithmetic sequence with common difference d. The sum S n of these n terms is given by where a n = a 1 + (n – 1)d.

9. Slide 11.2- 9 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 4 Finding the Sum of Terms of a Finite Arithmetic Sequence Find the sum of the arithmetic sequence of numbers: 1 + 4 + 7 + … + 25 Solution Arithmetic sequence with a 1 = 1 and d = 3. First find the number of terms.

10. Slide 11.2- 10 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 4 Finding the Sum of Terms of a Finite Arithmetic Sequence Solution continued Thus 1 + 4 + 7 + … + 25 = 117.