Copyright © 2009 Pearson Addison-Wesley. All rights reserved.
1 © 2010 Pearson Education, Inc. All rights reserved © 2010 Pearson Education, Inc. All rights...
-
Upload
reynard-bradley -
Category
Documents
-
view
228 -
download
3
Transcript of 1 © 2010 Pearson Education, Inc. All rights reserved © 2010 Pearson Education, Inc. All rights...
1© 2010 Pearson Education, Inc. All rights reserved
© 2010 Pearson Education, Inc.
All rights reserved
Chapter 10
Further Topics
in Algebra
© 2010 Pearson Education, Inc. All rights reserved
OBJECTIVES
2
Geometric Sequences and SeriesSECTION 10.3
1
2
Identify a geometric sequence and find its common ratio.Find the sum of a finite geometric sequence.Solve annuity problems.Find the sum of an infinite geometric sequence.
3
4
3© 2010 Pearson Education, Inc. All rights reserved
DEFINITION OF A GEOMETRIC SEQUENCEThe sequence a
1, a
2,
a3, a
4, … , a
n, …
is a geometric sequence, or a geometric progression, if there is a number r
such that each term except the first in the sequence is obtained by multiplying
the previous term by r. The number r is called the common ratio of the
geometric sequence.
1 , 1n
n
ar n
a
4© 2010 Pearson Education, Inc. All rights reserved
RECURSIVE DEFINITION OF A GEOMETRIC SEQUENCE
A geometric sequence
a1, a
2, a
3, a
4, … , a
n, …
can be defined recursively. The recursive formula
an + 1
= ran, n ≥ 1
defines a geometric sequence with first term a1 and common ratio r.
5© 2010 Pearson Education, Inc. All rights reserved
THE GENERAL TERM OF A GEOMETRIC SEQUENCE
Every geometric sequence can be written in the form
a1, a
1r, a
1r2, a
1r3, … , a
1rn−1, …
where r is the common ratio. Since a1 = a
1(1) = a
1r0, the nth term of
the geometric sequence is
an
= a1rn–1, for n ≥ 1.
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 Terms in a Geometric Sequence
For the geometric sequence 1, 3, 9, 27, …, find each of the following:
a. a1
b. r c. an
r 3
13.
Solution
a. The first term of the sequence is given a1 = 1.
b. Find the ratio of any two consecutive terms:
11
1 11 33 n n nn raa c.
9© 2010 Pearson Education, Inc. All rights reserved
Practice Problem
10© 2010 Pearson Education, Inc. All rights reserved
EXAMPLE 4 Finding a Particular Term in a Geometric Sequence
Find the 23rd term of a geometric sequence whose first term is 10 and whose
common
ratio is 1.2.
Solution
11© 2010 Pearson Education, Inc. All rights reserved
Practice Problem
12© 2010 Pearson Education, Inc. All rights reserved
SUM OF THE TERMS OF A FINITE GEOMETRIC SEQUENCE
Let a1, a
2, a
3, … a
n be the first n terms of a geometric sequence with first
term a1 and common ration r. The sum S
n of these terms is
111
1
1, 1.
1
nni
ni
a rS a r r
r
13© 2010 Pearson Education, Inc. All rights reserved
EXAMPLE 5 Finding the Sum of Terms of a Finite Geometric Sequence
Find each sum.
a. a1 = 5, r = 0.7, n = 15
Solution
15 15
1
1 1
5 0.7 5 0.7i i
i i
a. b.
1515
115 1
1
1 0.75 16.588
1 0.7i
i
S a r
15 15
1
1 1
5 0.7 0.7 5 0.7i i
i i
b. 0.7 16.588
11.6116
14© 2010 Pearson Education, Inc. All rights reserved
Practice Problem
15© 2010 Pearson Education, Inc. All rights reserved
16© 2010 Pearson Education, Inc. All rights reserved
17© 2010 Pearson Education, Inc. All rights reserved
18© 2010 Pearson Education, Inc. All rights reserved
VALUE OF AN ANNUITYLet P represent the payment in dollars made at the end of each of n
compounding periods per year, and let i be the annual interest rate. Then the
value of A of the annuity after t years is:
1 1nt
i
nA P
i
n
19© 2010 Pearson Education, Inc. All rights reserved
EXAMPLE 6 Finding the Value of an Annuity
An individual retirement account (IRA) is a common way to save money to
provide funds after retirement. Suppose you make payments of $1200 into an IRA
at the end of each year at an annual interest rate of 4.5% per year, compounded
annually. What is the value of this annuity after 35 years?
20© 2010 Pearson Education, Inc. All rights reserved
EXAMPLE 6 Finding the Value of an Annuity
Solution
351
1 11
1$97,795
1200
0.045
94
0.045
.
A
P = $1200, i = 0.045 and t = 35 years
The value of the IRA after 35 years is $97,795.94.
21© 2010 Pearson Education, Inc. All rights reserved
Practice Problem
22© 2010 Pearson Education, Inc. All rights reserved
Practice Problem
23© 2010 Pearson Education, Inc. All rights reserved
Practice Problem
24© 2010 Pearson Education, Inc. All rights reserved
SUM OF THE TERMS OF AN INFINITE GEOMETRIC SEQUENCE
If |r| < 1, the infinite sum
a1 + a
1r + a
1r2 + a
1r3 + … + a
1rn–1 + …
is given by
S a1ri 1
i1
a1
1 r.
25© 2010 Pearson Education, Inc. All rights reserved
EXAMPLE 7 Finding the Sum of an Infinite Geometric Series
Find the sum
Since |r| < 1, use the formula:
Solution
a1 2 and r
322
3
4
3 9 272 .
2 8 32
1 82
41 3
1rS
a
26© 2010 Pearson Education, Inc. All rights reserved
Practice Problem
27© 2010 Pearson Education, Inc. All rights reserved
Practice Problem