1 © 2010 Pearson Education, Inc. All rights reserved © 2010 Pearson Education, Inc. All rights reserved Chapter 10 Further Topics in Algebra

OBJECTIVES © 2010 Pearson Education, Inc. All rights reserved 2 Arithmetic Sequences; Partial Sums SECTION Identify an arithmetic sequence and find its common difference. Find the sum of the first n terms of an arithmetic sequence.

3 © 2010 Pearson Education, Inc. All rights reserved 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 © 2010 Pearson Education, Inc. All rights reserved 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 © 2010 Pearson Education, Inc. All rights reserved 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 © 2010 Pearson Education, Inc. All rights reserved Practice Problem

7 © 2010 Pearson Education, Inc. All rights reserved Practice Problem

8 © 2010 Pearson Education, Inc. All rights reserved 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

9 © 2010 Pearson Education, Inc. All rights reserved 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

10 © 2010 Pearson Education, Inc. All rights reserved Practice Problem

11 © 2010 Pearson Education, Inc. All rights reserved 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.

12 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 4 Finding the Sum of Terms of a Finite Arithmetic Sequence Find the sum of the arithmetic sequence of numbers: … + 25 Solution Arithmetic sequence with a 1 = 1 and d = 3. First find the number of terms.

13 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 4 Finding the Sum of Terms of a Finite Arithmetic Sequence Solution continued Thus … + 25 = 117.

14 © 2010 Pearson Education, Inc. All rights reserved Practice Problem

15 © 2010 Pearson Education, Inc. All rights reserved Practice Problem