Transcript Section 10.3 - University of South Florida
Chapter 10 Further Topics in Algebra
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SECTION 10.3
Geometric Sequences and Series
OBJECTIVES 1 2 3 4 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.
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DEFINITION OF A GEOMETRIC SEQUENCE The sequence
a
1 ,
a
2 ,
a
3 ,
a
4 , … ,
a n , …
is a
geometric sequence
, or a
geometric progression,
r
if there is a number each term except the first in the sequence is obtained by multiplying the previous term by . The number
r
is called the
r
such that
common ratio
of the geometric sequence.
a n
1
r
,
a n n
1
3
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RECURSIVE DEFINITION OF A GEOMETRIC SEQUENCE A geometric sequence formula
a a
1
n
,
a
+ 1 2 , =
a
3
ra
,
n a
4 , , … ,
n
≥ 1
a n , …
can be defined recursively. The recursive defines a geometric sequence with first term
a
1 and common ratio
r
.
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THE GENERAL TERM OF A GEOMETRIC SEQUENCE Every geometric sequence can be written in the form
a
1 ,
a
1
r
,
a
1
r
2 ,
a
1
r
3 ,
… , a
1
r n
− 1 , … where
r
is the common ratio. Since
a
1 =
a
1 (1) =
a
1
r
0 , the
n
th term of the geometric sequence
is
a n
=
a
1
r n
–1 , for
n
≥ 1.
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Practice Problem © 2010 Pearson Education, Inc. All rights reserved
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Practice Problem © 2010 Pearson Education, Inc. All rights reserved
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EXAMPLE 3 Finding Terms in a Geometric Sequence For the geometric sequence 1, 3, 9, 27, …, find each of the following: a. a 1 b. r c. a
n
Solution
a.
The first term of the sequence is given
a
1 = 1.
b.
Find the ratio of any two consecutive terms:
c.
a n
a
1
r n
1
r
3
n
3 1 3.
3
n
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Practice Problem © 2010 Pearson Education, Inc. All rights reserved
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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.
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Practice Problem © 2010 Pearson Education, Inc. All rights reserved
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SUM OF THE TERMS OF A FINITE GEOMETRIC SEQUENCE Let
a
1 ,
a
2 ,
a
3 ,
… a n
be the first
n
terms of a geometric sequence with first term
a
1 and common ration
r
. The sum
S n
is of these terms
S n
i n
1
a r
1
i
1
a
1 1 1
r r n
,
r
1.
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EXAMPLE 5 Finding the Sum of Terms of a Finite Geometric Sequence Find each sum.
a.
i
15 1 Solution
i
1
b.
i
15 1 a. a 1 = 5,
r S
15
i
15 1 = 0.7,
n a r
1
i
1 = 15 5 1
15 16.588
b.
i
15 1
i
15 1
i
1 11.6116
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i
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Practice Problem © 2010 Pearson Education, Inc. All rights reserved
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VALUE OF AN ANNUITY Let
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
years is:
A
P
of the annuity after 1
i n i n
nt
1
t
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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?
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EXAMPLE 6 Finding the Value of an Annuity Solution
P
= $1200 ,
i A
= 0.045
1200 and 1
t
= 35 0.045
1 years 35 1 0.045
1 $97,795 .
94 The value of the IRA after 35 years is $97,795.94.
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Practice Problem © 2010 Pearson Education, Inc. All rights reserved
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Practice Problem © 2010 Pearson Education, Inc. All rights reserved
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Practice Problem © 2010 Pearson Education, Inc. All rights reserved
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SUM OF THE TERMS OF AN INFINITE GEOMETRIC SEQUENCE If |
r
| < 1, the infinite sum
a
1 +
a
1
r
+
a
1
r
2 +
a
1
r
3 +
…
+
a
1
r n
–1 + … is given by
S
i
1
a
1
r i
1
a
1 1
r
.
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EXAMPLE 7 Finding the Sum of an Infinite Geometric Series Find the sum 2 2 9 8 27 32 .
Solution
a
1 2 and
r
3 2 2 3 4 Since |
r
| < 1, use the formula:
S
1
a
1
r
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Practice Problem © 2010 Pearson Education, Inc. All rights reserved
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Practice Problem © 2010 Pearson Education, Inc. All rights reserved
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