chapter 8 sequences, induction, and probability copyright © 2014, 2010, 2007 pearson education,...

Chapter 8Sequences,

Induction, and

Probability

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 1

8.2 Arithmetic Sequences


• 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.

Objectives:


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.

1 100a

2 1 30 100 30 70a a

3 2 30 70 30 40a a

4 3 30 40 30 10a a

5 4 30 10 30 20a a

6 5 30 20 30 50a a

The terms are100, 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

1 ( 1)na a n d


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.

1 ( 1)na a n d 9 1 (9 1)

6 (9 1)( 5)

a a d

6 8( 5)

6 40 34


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. Population (continued)

1 ( 1)na a n d

1 16a

0.35d

1 ( 1) 16 0.35( 1)na a n d n

16 0.35 0.35 0.35 15.65n n

The formula for the nth term of the arithmetic sequence thatdescribes the percentage of the U.S. population that will beLatino n years after 2009 is 0.35 15.65.na n


Example: Using an Arithmetic Sequence to Model Changes in the U.S. Population (continued)

We have found that the formulafor the nth term of the arithmetic sequence thatdescribes the percentage of the U.S. population that will beLatino n years after 2009 is

0.35 15.65.na n

What percentage of the U.S. population is projected to beLatino in 2030? 2030 2009 21n

21 0.35(21) 15.65 23a 23% of the U.S. populationis 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.

1( ),2n n

nS a a


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.

1( ),2n n

nS a a

15

15 15(3 45) (48) 360

2 2S

1 ( 1)na a n d

15 3 (15 1)3 3 14(3) 45a The sum of the first15 terms of thesequence is 360.

chapter 8 sequences, induction, and probability copyright © 2014, 2010, 2007 pearson education,...

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