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Sequences,
Induction, and
Probability
10
Something incredible has happened.
Your college roommate, a gifted athlete,
has been given a six-year contract with a
professional baseball team. He will be playing
against the likes of Alex Rodriguez and Manny
Ramirez. Management offers him three options. One
is a beginning salary of $1,700,000 with annual increases
of $70,000 per year starting in the second year. A second
option is $1,700,000 the first year with an annual increase of
2% per year beginning in the second year. The third option
involves less money the first year—$1,500,000—but there is an
annual increase of 9% yearly after that. Which option offers the
most money over the six-year contract?
A similar problem appears as Exercise 67 in Exercise Set 10.3, and this
problem appears as the Group Exercise on page 986.
951
951
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952 Chapter 10 Sequences, Induction, and Probability
Section
10.1 Sequences and Summation Notation
Objectives
Sequences
� Find particular terms of a
�
�
�
M
any creations in nature involve intricate mathematical designs, including a variety of spirals. For
example, the arrangement of the individual florets in
the head of a sunflower forms spirals. In some species,
there are 21 spirals in the clockwise direction and 34 in
the counterclockwise direction. The precise numbers
depend on the species of sunflower: 21 and 34, or 34 and
55, or 55 and 89, or even 89 and 144.
This observation becomes even more interesting
when we consider a sequence of numbers investigated by
Leonardo of Pisa, also known as Fibonacci, an Italian
mathematician of the thirteenth century. The Fibonacci
sequence of numbers is an infinite sequence that begins as
follows:
sequence from the general
term.
Use recursion formulas.
Use factorial notation.
Use summation notation.
Fibonacci Numbers
on the Piano
Keyboard
One Octave
Numbers in the Fibonacci
sequence can be found in an
octave on the piano keyboard. The
octave contains 2 black keys in one
cluster and 3 black keys in another
cluster, for a total of 5 black keys. It
also has 8 white keys, for a total of
13 keys. The numbers 2, 3, 5, 8, and
13 are the third through seventh
terms of the Fibonacci sequence.
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, Á .
The first two terms are 1. Every term thereafter is the sum of the two preceding
terms. For example, the third term, 2, is the sum of the first and second terms:
1 + 1 = 2. The fourth term, 3, is the sum of the second and third terms: 1 + 2 = 3,
and so on. Did you know that the number of spirals in a daisy or a sunflower, 21 and
34, are two Fibonacci numbers? The number of spirals in a pine cone, 8 and 13, and
a pineapple, 8 and 13, are also Fibonacci numbers.
We can think of the Fibonacci sequence as a function.The terms of the sequence
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, Á
are the range values for a function f whose domain is the set of positive integers.
Domain:
1,
T
1,
Range:
2,
T
1,
3,
T
2,
4,
T
3,
5,
T
5,
6,
T
8,
7,
T
13,
Á
Á
Thus, f112 = 1, f122 = 1, f132 = 2, f142 = 3, f152 = 5, f162 = 8, f172 = 13, and
so on.
The letter a with a subscript is used to represent function values of a
sequence, rather than the usual function notation. The subscripts make up the
domain of the sequence and they identify the location of a term. Thus, a1 represents the first term of the sequence, a2 represents the second term, a3 the third
term, and so on. This notation is shown for the first six terms of the Fibonacci
sequence:
1,
1,
2,
3,
5,
8.
a1 = 1
a2 = 1
a3 = 2
a4 = 3
a5 = 5
a6 = 8
The notation a n represents the nth term, or general term, of a sequence. The
entire sequence is represented by 5a n6.
Definition of a Sequence
An infinite sequence 5an6 is a function whose domain is the set of positive
integers. The function values, or terms, of the sequence are represented by
a1 , a2 , a3 , a4 , Á , an , Á .
Sequences whose domains consist only of the first n positive integers are called
finite sequences.
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Section 10.1 Sequences and Summation Notation
�
Find particular terms
of a sequence from
the general term.
953
Writing Terms of a Sequence from the General Term
EXAMPLE 1
Write the first four terms of the sequence whose nth term, or general term, is given:
a. a n = 3n + 4
b. an =
1- 12n
3n - 1
.
Solution
a. We need to find the first four terms of the sequence whose general term is
a n = 3n + 4. To do so, we replace n in the formula with 1, 2, 3, and 4.
a1, 1st
term
3 ⴢ 1+4=3+4=7
a2, 2nd
term
3 ⴢ 2+4=6+4=10
a3, 3rd
term
3 ⴢ 3+4=9+4=13
a4, 4th
term
3 ⴢ 4+4=12+4=16
The first four terms are 7, 10, 13, and 16. The sequence defined by an = 3n + 4
can be written as
7, 10, 13, 16, Á , 3n + 4, Á .
Study Tip
The factor 1 - 12 in the general term
of a sequence causes the signs of the
terms to alternate between positive
and negative, depending on whether
n is even or odd.
n
b. We need to find the first four terms of the sequence whose general term is
1- 12n
an = n
. To do so, we replace each occurrence of n in the formula with
3 - 1
1, 2, 3, and 4.
a1, 1st
term
–1
1
(–1)1
=
=–
3-1
2
31-1
a2, 2nd
term
1
1
(–1)2
=
=
9-1
8
32-1
a3, 3rd
term
–1
1
(–1)3
=
=–
3
27-1
26
3 -1
a4, 4th
term
1
1
(–1)4
=
=
4
81-1
80
3 -1
The first four terms are be written as
1 1
2, 8,
-
1
26 ,
and
1
80 . The
sequence defined by
1- 12n
3n - 1
can
1-12n
1 1
1 1
- , , - , ,Á, n
,Á.
2 8
26 80
3 - 1
Check Point
1
Write the first four terms of the sequence whose nth term, or
general term, is given:
a. a n = 2n + 5
b. a n =
1- 12n
2n + 1
.
Although sequences are usually named with the letter a, any lowercase letter
can be used. For example, the first four terms of the sequence 5bn6 = E A 12 B n F are
1
b1 = 12 , b2 = 14 , b3 = 18 , and b4 = 16
.
Because a sequence is a function whose domain is the set of positive integers,
the graph of a sequence is a set of discrete points. For example, consider the
sequence whose general term is an = n1 . How does the graph of this sequence differ
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954 Chapter 10 Sequences, Induction, and Probability
from the graph of the function f1x2 = x1 ? The graph of f1x2 =
Technology
Graphing utilities can write the terms
of a sequence and graph them. For
example, to find the first six terms
1
of 5an6 = e f , enter
n
General
term
Variable
used in
general
term
Start
at a1.
The first few terms of the sequence
are shown in the viewing rectangle. By
pressing the right arrow key to scroll
right, you can see the remaining terms.
E n1 F ,
remove all the points from the graph of f except those whose x-coordinates are positive
integers. Thus, we remove all points except 11, 12, A 2, 12 B , A 3, 13 B , A 4, 14 B , and so on. The
remaining points are the graph of the sequence 5an6 = E n1 F , shown in Figure 10.1(b).
Notice that the horizontal axis is labeled n and the vertical axis is labeled an .
y
The
“step”
from
a1 to a2,
a2 to a3,
etc., is 1.
is shown in
Figure 10.1(a) for positive values of x. To obtain the graph of the sequence 5a n6 =
Stop
at a6.
SEQ (1÷x, x, 1, 6, 1).
1
x
an
3
3
2
2
(1, 1)
1
1
(1, 1)
(2, q) (3, a) (4, ~)
2
3
4
1
x
(2, q) (3, a) (4, ~)
1
2
3
4
n
Figure 10.1(a) The graph of
Figure 10.1(b) The graph of
1
1
f1x2 = , x 7 0
5an6 = e f
x
n
Comparing a continuous graph to the graph of a sequence
Recursion Formulas
In Example 1, the formulas used for the nth term of a sequence expressed the term as
a function of n, the number of the term. Sequences can also be defined using recursion
formulas. A recursion formula defines the nth term of a sequence as a function of the
previous term. Our next example illustrates that if the first term of a sequence is
known, then the recursion formula can be used to determine the remaining terms.
�
Use recursion formulas.
EXAMPLE 2
Using a Recursion Formula
Find the first four terms of the sequence in which a1 = 5 and an = 3an - 1 + 2 for
n Ú 2.
Solution Let’s be sure we understand what is given.
a1=5
The first
term is 5.
and
an
Each term
after the first
=
is
3an-1 +2
3 times the
previous term
plus 2.
Now let’s write the first four terms of this sequence.
a1 = 5
This is the given first term.
a2 = 3a1 + 2
Use an = 3an - 1 + 2, with n = 2.
Thus, a2 = 3a2 - 1 + 2 = 3a1 + 2.
= 3152 + 2 = 17
a3 = 3a2 + 2
= 31172 + 2 = 53
a4 = 3a3 + 2
= 31532 + 2 = 161
Substitute 5 for a1 .
Again use an = 3an - 1 + 2, with n = 3.
Substitute 17 for a2 .
Notice that a4 is defined in terms of a3 .
We used an = 3an - 1 + 2, with n = 4.
Use the value of a3 , the third term,
obtained above.
The first four terms are 5, 17, 53, and 161.
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Section 10.1 Sequences and Summation Notation
Check Point
955
2
Find the first four terms of the sequence in which a 1 = 3 and
an = 2an - 1 + 5 for n Ú 2.
�
Use factorial notation.
Factorial Notation
Products of consecutive positive integers occur quite often in sequences. These
products can be expressed in a special notation, called factorial notation.
Factorials from 0
through 20
0!
1!
2!
3!
4!
5!
6!
7!
8!
9!
10!
11!
12!
13!
14!
15!
16!
17!
18!
19!
20!
1
1
2
6
24
120
720
5040
40,320
362,880
3,628,800
39,916,800
479,001,600
6,227,020,800
87,178,291,200
1,307,674,368,000
20,922,789,888,000
355,687,428,096,000
6,402,373,705,728,000
121,645,100,408,832,000
2,432,902,008,176,640,000
As n increases, n! grows very
rapidly. Factorial growth is more
explosive than exponential growth
discussed in Chapter 3.
Factorial Notation
If n is a positive integer, the notation n! (read “n factorial”) is the product of all
positive integers from n down through 1.
n! = n1n - 121n - 22 Á 132122112
0! (zero factorial), by definition, is 1.
0! = 1
The values of n! for the first six positive integers are
1! = 1
2! = 2 # 1 = 2
3! = 3 # 2 # 1 = 6
4! = 4 # 3 # 2 # 1 = 24
5! = 5 # 4 # 3 # 2 # 1 = 120
6! = 6 # 5 # 4 # 3 # 2 # 1 = 720.
Factorials affect only the number or variable that they follow unless grouping
symbols appear. For example,
2 # 3! = 213 # 2 # 12 = 2 # 6 = 12
whereas
12 # 32! = 6! = 6 # 5 # 4 # 3 # 2 # 1 = 720.
In this sense, factorials are similar to exponents.
EXAMPLE 3
Finding Terms of a Sequence Involving Factorials
Write the first four terms of the sequence whose nth term is
an =
Technology
2n
.
1n - 12!
Solution We need to find the first four terms of the sequence. To do so, we replace
Most calculators have factorial keys.
To find 5!, most calculators use one
of the following:
Many Scientific Calculators
5冷x! 冷
Many Graphing Calculators
5冷! 冷 冷ENTER 冷.
Because n! becomes quite large as n
increases, your calculator will
display these larger values in
scientific notation.
each n in
2n
with 1, 2, 3, and 4.
1n - 12!
a1, 1st
term
2
2
21
= = =2
0!
1
(1-1)!
a2, 2nd
term
4
4
22
= = =4
(2-1)!
1!
1
a3, 3rd
term
23
8
8
= =
=4
(3-1)!
2!
2ⴢ1
a4, 4th
term
16
16
16
8
24
= =
= =
6
(4-1)!
3!
3ⴢ2ⴢ1
3
The first four terms are 2, 4, 4, and 83 .
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956 Chapter 10 Sequences, Induction, and Probability
3
Check Point
Write the first four terms of the sequence whose nth term is
20
an =
.
1n + 12!
When evaluating fractions with factorials in the numerator and the denominator,
try to reduce the fraction before performing the multiplications. For example, consider
26!
. Rather than write out 26! as the product of all integers from 26 down to 1, we can
21!
express 26! as
26! = 26 # 25 # 24 # 23 # 22 # 21!.
In this way, we can divide both the numerator and the denominator by the common
factor, 21!.
26!
26 # 25 # 24 # 23 # 22 # 21!
=
= 26 # 25 # 24 # 23 # 22 = 7,893,600
21!
21!
Evaluating Fractions with Factorials
EXAMPLE 4
Evaluate each factorial expression:
a.
10!
2!8!
b.
1n + 12!
n!
.
Solution
10!
10 # 9 # 8!
90
= # #
=
= 45
2!8!
2 1 8!
2
1n + 12!
1n + 12 # n!
b.
=
= n + 1
n!
n!
a.
Check Point
a.
�
Use summation notation.
14!
2!12!
4
Evaluate each factorial expression:
b.
n!
.
1n - 12!
Summation Notation
It is sometimes useful to find the sum of the first n terms of a sequence. For example,
consider the cost of raising a child born in the United States in 2006 to a middleincome ($43,200–$72,600 per year) family, shown in Table 10.1.
Table 10.1 The Cost of Raising a Child Born in the U.S. in 2006 to a Middle-Income Family
Year
Average Cost
2006
Average Cost
2008
2009
2010
2011
2012
2013
2014
$10,600 $10,930 $11,270 $11,960 $12,330 $12,710 $12,950 $13,350 $13,760
Child
is under 1.
Year
2007
Child
is 1.
Child
is 2.
Child
is 3.
Child
is 4.
Child
is 5.
Child
is 6.
Child
is 7.
Child
is 8.
2016
2017
2018
2019
2020
2021
2022
2023
2015
$13,970 $14,400 $14,840 $16,360 $16,860 $17,390 $18,430 $19,000 $19,590
Child
is 9.
Child
is 10.
Source: U.S. Department of Agriculture
Child
is 11.
Child
is 12.
Child
is 13.
Child
is 14.
Child
is 15.
Child
is 16.
Child
is 17.
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Section 10.1 Sequences and Summation Notation
957
We can let an represent the cost of raising a child in year n, where n = 1
corresponds to 2006, n = 2 to 2007, n = 3 to 2008, and so on. The terms of the finite
sequence in Table 10.1 are given as follows:
10,600, 10,930, 11,270, 11,960, 12,330, 12,710, 12,950, 13,350, 13,760,
a1
a2
a3
a4
a5
a6
a7
a8
a9
13,970, 14,400, 14,840, 16,360, 16,860, 17,390, 18,430, 19,000, 19,590.
a10
a11
a12
a13
a14
a15
a16
a17
a18
Why might we want to add the terms of this sequence? We do this to find the
total cost of raising a child born in 2006 from birth through age 17. Thus,
a1 + a2 + a3 + a4 + a5 + a6 + a7 + a8 + a9 + a10 + a11 + a12 + a13 + a14 + a15 + a16 + a17 + a18
= 10,600 + 10,930 + 11,270 + 11,960 + 12,330 + 12,710 + 12,950 + 13,350 + 13,760
+ 13,970 + 14,400 + 14,840 + 16,360 + 16,860 + 17,390 + 18,430 + 19,000 + 19,590
= 260,700.
We see that the total cost of raising a child born in 2006 from birth through age 17 is
$260,700.
There is a compact notation for expressing the sum of the first n terms of a
sequence. For example, rather than write
a1 + a2 + a3 + a4 + a5 + a6 + a7 + a8 + a9 + a10 + a11 + a12 + a13 + a14 + a15 + a16 + a17 + a18 ,
we can use summation notation to express the sum as
18
a ai .
i=1
We read this expression as “the sum as i goes from 1 to 18 of ai .” The letter i is called
the index of summation and is not related to the use of i to represent 2-1.
You can think of the symbol © (the uppercase Greek letter sigma) as an
instruction to add up the terms of a sequence.
Summation Notation
The sum of the first n terms of a sequence is represented by the summation notation
n
Á + an ,
a ai = a1 + a2 + a3 + a4 +
i=1
where i is the index of summation, n is the upper limit of summation, and 1 is the
lower limit of summation.
Any letter can be used for the index of summation.The letters i, j, and k are used
commonly. Furthermore, the lower limit of summation can be an integer other than 1.
When we write out a sum that is given in summation notation, we are
expanding the summation notation. Example 5 shows how to do this.
EXAMPLE 5
Using Summation Notation
Expand and evaluate the sum:
6
a. a 1i2 + 12
i=1
7
b. a 31 -22k - 54
k=4
5
c. a 3.
i=1
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958 Chapter 10 Sequences, Induction, and Probability
Solution
Technology
6
Graphing utilities can calculate the
sum of a sequence. For example, to
find the sum of the sequence in
Example 5(a), enter
冷SUM 冷 冷SEQ 冷 1x2
a. To find a 1i2 + 12, we must replace i in the expression i2 + 1 with all consecutive
i=1
integers from 1 to 6, inclusive.Then we add.
6
2
2
2
2
2
a 1i + 12 = 11 + 12 + 12 + 12 + 13 + 12 + 14 + 12
i=1
+ 1, x, 1, 6, 12.
+ 152 + 12 + 162 + 12
Then press 冷ENTER 冷; 97 should be
displayed. Use this capability to verify
Example 5(b).
= 2 + 5 + 10 + 17 + 26 + 37
= 97
7
b. The index of summation in a 31 -22k - 54 is k. First we evaluate 1 -22k - 5
k=4
for all consecutive integers from 4 through 7, inclusive. Then we add.
7
k
4
5
a 31 -22 - 54 = 31 - 22 - 54 + 31 -22 - 54
k=4
+ 31 -226 - 54 + 31 - 227 - 54
= 116 - 52 + 1-32 - 52 + 164 - 52 + 1-128 - 52
= 11 + 1-372 + 59 + 1 -1332
= - 100
5
c. To find a 3, we observe that every term of the sum is 3. The notation i = 1
i=1
through 5 indicates that we must add the first five terms of a sequence in which
every term is 3.
5
a 3 = 3 + 3 + 3 + 3 + 3 = 15
i=1
Check Point
6
5
Expand and evaluate the sum:
5
b. a 12 k - 32
a. a 2i2
i=1
k=3
5
c. a 4.
i=1
Although the domain of a sequence is the set of positive integers, any
integers can be used for the limits of summation. For a given sum, we can vary
the upper and lower limits of summation, as well as the letter used for the index
of summation. By doing so, we can produce different-looking summation
notations for the same sum. For example, the sum of the squares of the first four
positive integers, 12 + 2 2 + 32 + 4 2, can be expressed in a number of equivalent
ways:
4
2
2
2
2
2
a i = 1 + 2 + 3 + 4 = 30
i=1
3
2
2
2
2
2
a 1i + 12 = 10 + 12 + 11 + 12 + 12 + 12 + 13 + 12
i=0
= 12 + 2 2 + 32 + 4 2 = 30
5
2
2
2
2
2
a 1k - 12 = 12 - 12 + 13 - 12 + 14 - 12 + 15 - 12
k=2
= 12 + 2 2 + 32 + 4 2 = 30.
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Section 10.1 Sequences and Summation Notation
959
Writing Sums in Summation Notation
EXAMPLE 6
Express each sum using summation notation:
1
1
1
1
a. 13 + 2 3 + 33 + Á + 73
b. 1 + + +
+ Á + n-1 .
3
9
27
3
Solution In each case, we will use 1 as the lower limit of summation and i for the
index of summation.
a. The sum 13 + 2 3 + 33 + Á + 73 has seven terms, each of the form i3, starting
at i = 1 and ending at i = 7. Thus,
7
13 + 2 3 + 33 + Á + 73 = a i3.
i=1
b. The sum
1
1
1
1
+ +
+ Á + n-1
3
9
27
3
1
has n terms, each of the form i - 1 , starting at i = 1 and ending at i = n. Thus,
3
n
1
1
1
1
1
1 + + +
+ Á + n-1 = a i-1 .
3
9
27
3
3
i=1
1 +
Check Point
6
Express each sum using summation notation:
a. 12 + 2 2 + 32 + Á + 92
b. 1 +
1
1
1
1
+ + + Á + n-1 .
2
4
8
2
Table 10.2 contains some important properties of sums expressed in summation
notation.
Table 10.2 Properties of Sums
Property
n
Example
n
1. a cai = c a ai , c any real number
i=1
i=1
4
2
# 2 # 2 # 2 # 2
a 3i = 3 1 + 3 2 + 3 3 + 3 4
i=1
4
3 a i2 = 3112 + 2 2 + 32 + 4 22 = 3 # 12 + 3 # 2 2 + 3 # 32 + 3 # 4 2
i=1
4
4
Conclusion: a 3i2 = 3 a i2
i=1
n
n
n
i=1
i=1
i=1
2. a 1ai + bi2 = a ai + a bi
i=1
4
2
2
2
2
2
a 1i + i 2 = 11 + 1 2 + 12 + 2 2 + 13 + 3 2 + 14 + 4 2
i=1
4
4
2
2
2
2
2
a i + a i = 11 + 2 + 3 + 42 + 11 + 2 + 3 + 4 2
i=1
i=1
= 11 + 122 + 12 + 2 22 + 13 + 322 + 14 + 4 22
4
4
4
i=1
i=1
i=1
Conclusion: a 1i + i22 = a i + a i2
n
n
n
i=1
i=1
i=1
3. a 1ai - bi2 = a ai - a bi
5
2
3
2
3
2
3
2
3
a 1i - i 2 = 13 - 3 2 + 14 - 4 2 + 15 - 5 2
i=3
5
5
i=3
i=3
2
3
2
2
2
3
3
3
a i - a i = 13 + 4 + 5 2 - 13 + 4 + 5 2
5
= 132 - 332 + 14 2 - 4 32 + 152 - 532
5
5
i=3
i=3
Conclusion: a 1i2 - i32 = a i2 - a i3
i=3
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960 Chapter 10 Sequences, Induction, and Probability
Exercise Set 10.1
Practice Exercises
In Exercises 1–12, write the first four terms of each sequence whose
general term is given.
1. an = 3n + 2
3. an = 3n
5. an = 1 -32n
7. an = 1 -12n1n + 32
2n
9. an =
n + 4
11. a n =
1 - 12n + 1
2n - 1
2. an = 4n - 1
1
4. an = a b
3
n
1
6. an = a - b
3
n
8. an = 1-12n + 11n + 42
3n
10. a n =
n + 5
12. an =
1- 12n + 1
46. 5 + 52 + 53 + Á + 512
47. 1 + 2 + 3 + Á + 30
48. 1 + 2 + 3 + Á + 40
49.
2
3
14
1
+ + + Á +
2
3
4
14 + 1
50.
2
3
16
1
+ + + Á +
3
4
5
16 + 2
51. 4 +
14. a1 = 12 and an = an - 1 + 4 for n Ú 2
15. a1 = 3 and an = 4an - 1 for n Ú 2
16. a1 = 2 and an = 5an - 1 for n Ú 2
17. a1 = 4 and an = 2an - 1 + 3 for n Ú 2
18. a1 = 5 and a n = 3an - 1 - 1 for n Ú 2
In Exercises 19–22, the general term of a sequence is given and
involves a factorial. Write the first four terms of each sequence.
1n + 12!
n2
19. an =
20. an =
n!
n2
21. an = 21n + 12!
22. an = - 21n - 12!
In Exercises 23–28, evaluate each factorial expression.
2
3
n
1
+ 2 + 3 + Á + n
9
9
9
9
53. 1 + 3 + 5 + Á + 12n - 12
54. a + ar + ar2 + Á + arn - 1
18!
16!
20!
26.
2!18!
12n + 12!
28.
12n2!
24.
In Exercises 55–60, express each sum using summation notation.
Use a lower limit of summation of your choice and k for the index
of summation.
55. 5 + 7 + 9 + 11 + Á + 31
56. 6 + 8 + 10 + 12 + Á + 32
57. a + ar + ar2 + Á + ar12
58. a + ar + ar2 + Á + ar14
59. a + 1a + d2 + 1a + 2d2 + Á + 1a + nd2
60. 1a + d2 + 1a + d22 + Á + 1a + dn2
Practice Plus
In Exercises 61–68, use the graphs of 5an6 and 5bn6 to find each
indicated sum.
In Exercises 29–42, find each indicated sum.
6
6
29. a 5i
30. a 7i
i=1
i=1
4
5
31. a 2i2
i=1
5
32. a i3
i=1
4
33. a k1k + 42
34. a 1k - 321k + 22
4
1 i
35. a a - b
2
4
1 i
36. a a - b
3
k=1
i=1
9
i=5
4
39. a
i=0
5
1 - 12i
i!
i!
41. a
i = 1 1i - 12!
The Graph of {an}
an
The Graph of {bn}
bn
5
4
3
2
1
5
4
3
2
1
−1
−2
−3
−4
−5
k=1
i=2
7
37. a 11
42
43
4n
+
+ Á +
n
2
3
52.
13. a1 = 7 and an = an - 1 + 5 for n Ú 2
17!
15!
16!
25.
2!14!
1n + 22!
27.
n!
44. 14 + 2 4 + 34 + Á + 12 4
45. 2 + 2 2 + 2 3 + Á + 2 11
2n + 1
The sequences in Exercises 13–18 are defined using recursion
formulas. Write the first four terms of each sequence.
23.
In Exercises 43–54, express each sum using summation notation.
Use 1 as the lower limit of summation and i for the index of
summation.
43. 12 + 2 2 + 32 + Á + 152
1 2 3 4 5
n
−1
−2
−3
−4
−5
1 2 3 4 5
38. a 12
i=3
4 1- 12i + 1
40. a
i = 0 1i + 12!
5
42. a
i=1
1i + 22!
i!
5
61. a 1a2i + 12
i=1
5
63. a 12ai + bi2
i=1
5
62. a 1b2i - 12
i=1
5
64. a 1ai + 3bi2
i=1
n
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Section 10.1 Sequences and Summation Notation
5
ai 2
65. a ¢ ≤
i = 4 bi
5
5
i=1
Let an represent spending for consumer drug ads, in billions of
dollars, n years after 2001.
5
ai 3
66. a ¢ ≤
i = 4 bi
5
67. a a2i + a b2i
a. Use the numbers given in the graph to find and interpret
1 5
ai .
5 ia
=1
5
68. a a2i - a b2i
i=1
i=1
i=3
b. The finite sequence whose general term is
an = 0.65n + 2.3,
Application Exercises
69. Advertisers don’t have to fear that they’ll face a sea of “sold
out” signs as they rush to the Internet. The growing number of
popular sites filled with user-created content, including
MySpace.com and YouTube.com, provide plenty of inventory
for advertisers who can’t find space on top portals such as
Yahoo. The bar graph shows U.S. online ad spending, in
billions of dollars, from 2000 through 2006.
Online Ad Spending
(billions of dollars)
United States Online Ad Spending
$18
$16
$14
$12
$10
$8
$6
$4
$2
16.7
where n = 1, 2, 3, 4, 5, models spending for consumer
drug ads, in billions of dollars, n years after 2001. Use the
1 5
model to find a ai .Does this seem reasonable in terms of
5 i=1
the actual sum in part (a), or has model breakdown
occurred?
71. A deposit of $6000 is made in an account that earns 6%
interest compounded quarterly. The balance in the account
after n quarters is given by the sequence
13.1
an = 6000a 1 +
10.0
8.1
8.1
7.2
2000 2001 2002 2003 2004 2005 2006
Year
a n = 10,000a 1 +
Let an represent online ad spending, in billions of dollars, n years
after 1999.
a. Use the numbers given in the graph to find and interpret
1 7
ai .
7 ia
=1
2
an = 0.5n - 1.5n + 8,
where n = 1, 2, 3, Á , 7, models online ad spending, an , in
billions of dollars, n years after 1999. Use the model to find
1 7
ai . Does this underestimate or overestimate the actual
7 ia
=1
sum in part (a)? By how much?
70. More and more television commercial time is devoted to
drug companies as hucksters for the benefits and risks of
their wares. The bar graph shows the amount that drug
companies spent on consumer drug ads, in billions of dollars,
from 2002 through 2006.
Spending for Consumer Drug Ads
Spending for Drug Ads
(billions of dollars)
$6
$3
0.08 n
b ,
4
n = 1, 2, 3, Á .
Find the balance in the account after six years. Round to the
nearest cent.
Writing in Mathematics
73. What is a sequence? Give an example with your description.
b. The finite sequence whose general term is
74. Explain how to write terms of a sequence if the formula for
the general term is given.
75. What does the graph of a sequence look like? How is it
obtained?
76. What is a recursion formula?
77. Explain how to find n! if n is a positive integer.
78. Explain the best way to evaluate
900!
without a calculator.
899!
79. What is the meaning of the symbol ©? Give an example with
your description.
80. You buy a new car for $24,000. At the end of n years, the
value of your car is given by the sequence
3 n
a n = 24,000a b ,
4
5.5
4.5
n = 1, 2, 3, Á .
72. A deposit of $10,000 is made in an account that earns 8%
interest compounded quarterly. The balance in the account
after n quarters is given by the sequence
Source: eMarketer
$4
0.06 n
b ,
4
Find the balance in the account after five years. Round to the
nearest cent.
6.1
$5
961
4.9
n = 1, 2, 3, Á .
Find a5 and write a sentence explaining what this value
represents. Describe the nth term of the sequence in terms of
the value of your car at the end of each year.
3.6
2.9
$2
Technology Exercises
$1
2002
2003
Source: Nielsen Monitor-Plus
2004
Year
2005
2006
In Exercises 81–85, use a calculator’s factorial key to evaluate each
expression.
81.
200!
198!
82. a
300
b!
20
83.
20!
300
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962 Chapter 10 Sequences, Induction, and Probability
84.
20!
120 - 32!
85.
54!
154 - 32!3!
86. Use the 冷SEQ 冷 (sequence) capability of a graphing utility to
verify the terms of the sequences you obtained for any five
sequences from Exercises 1–12 or 19–22.
87. Use the 冷SUM 冷 冷SEQ 冷 (sum of the sequence) capability of a
graphing utility to verify any five of the sums you obtained in
Exercises 29–42.
88. As n increases, the terms of the sequence
an = a 1 +
1 n
b
n
get closer and closer to the number e (where e L 2.7183). Use a
calculator to find a 10 , a100 , a1000 , a10,000 , and a100,000 , comparing
these terms to your calculator’s decimal approximation for e.
94. By writing a1 , a 2 , a3 , a4 , Á , an , Á , I can see that the
range of a sequence is the set of positive integers.
95. It makes a difference whether or not I use parentheses
around the expression following the summation symbol,
8
because the value of a 1i + 72 is 92, but the value of
8
i=1
96. Without writing out the terms, I can see that 1- 122n in
1- 122n
causes the terms to alternate in sign.
an =
3n
In Exercises 97–100, determine whether each statement is true or
false. If the statement is false, make the necessary change(s) to
produce a true statement.
97.
Many graphing utilities have a sequence-graphing mode that plots
the terms of a sequence as points on a rectangular coordinate system.
Consult your manual; if your graphing utility has this capability, use
it to graph each of the sequences in Exercises 89–92. What appears to
be happening to the terms of each sequence as n gets larger?
n
89. a n =
n + 1
90. an =
100
n
n:30, 10, 14 by an:30, 1, 0.14
1
n!
=
1n - 12!
n - 1
98. The Fibonacci sequence 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89,
144, Á can be defined recursively using a 0 = 1, a1 = 1;
an = an - 2 + an - 1 , where n Ú 2.
2
99. a 1- 12i2 i = 0
i=1
2
n:30, 1000, 1004 by an:30, 1, 0.14
2n + 5n - 7
n3
i=1
2
3n + n - 1
5n4 + 2n2 + 1
i=1
i=1
101. Write the first five terms of the sequence whose first term is
9 and whose general term is
n:30, 10, 14 by an:30, 2, 0.24
an - 1
an = c 2
3a n - 1 + 5
4
92. a n =
2
100. a aibi = a ai a bi
2
91. an =
i=1
a i + 7 is 43.
n:30, 10, 14 by an:30, 1, 0.14
if an - 1 is even
if a n - 1 is odd
for n Ú 2.
Critical Thinking Exercises
Group Exercise
Make Sense? In Exercises 93–96, determine whether each
statement makes sense or does not make sense, and explain
your reasoning.
102. Enough curiosities involving the Fibonacci sequence exist to
warrant a flourishing Fibonacci Association, which publishes
a quarterly journal. Do some research on the Fibonacci
sequence by consulting the Internet or the research department of your library, and find one property that interests you.
After doing this research, get together with your group to
share these intriguing properties.
93. Now that I’ve studied sequences, I realize that the joke in this
cartoon is based on the fact that you can’t have a negative
number of sheep.
Preview Exercises
Exercises 103–105 will help you prepare for the material covered
in the next section.
103. Consider the sequence 8, 3, -2, - 7, -12, Á . Find a2 - a1 ,
a 3 - a2 , a4 - a3 , and a5 - a4 . What do you observe?
104. Consider the sequence whose nth term is an = 4n - 3.
Find a 2 - a1 , a3 - a2 , a4 - a3 , and a5 - a4 . What do you
observe?
105. Use the formula a n = 4 + 1n - 121 - 72 to find the eighth
term of the sequence 4, -3, -10, Á .