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Slide 9- 1
Chapter 9
Discrete Mathematics
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9.1
Basic Combinatorics
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Quick Review
Give the number of objects described.
1. The number of cards in a standard deck.
2. The number of face cards in a standard deck.
3. The number of vertices of a octogon.
4. The number of faces on a cubical die.
5. The number of possible totals when two dice are rolled.
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Slide 9- 4
Quick Review Solutions
Give the number of objects described.
1. The number of cards in a standard deck. 52
2. The number of face cards in a standard deck. 12
3. The number of vertices of a octogon. 8
4. The number of faces on a cubical die. 6
5. The number of possible totals when two dice are rolled. 11
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Slide 9- 5
What you’ll learn about






Discrete Versus Continuous
The Importance of Counting
The Multiplication Principle of Counting
Permutations
Combinations
Subsets of an n-Set
… and why
Counting large sets is easy if you know the correct formula.
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Slide 9- 6
Multiplication Principle of Counting
If a procedure P has a sequence of stages S , S ,..., S and if
1
2
n
S can occur in r ways,
1
1
S can occur in r ways
2
2
S can occur in r ways,
n
n
then the number of ways that the procedure P can occur is the
product rr ...r .
1
2
n
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Slide 9- 7
Example Using the Multiplication
Principle
If a license plate has four letters followed by three numerical
digits. Find the number of different license plates that could be
formed if there is no restriction on the letters or digits that can be
used.
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Slide 9- 8
Example Using the Multiplication
Principle
If a license plate has four letters followed by three numerical
digits. Find the number of different license plates that could be
formed if there is no restriction on the letters or digits that can be
used.
You can fill in the first blank 26 ways, the second blank 26
ways, the third blank 26 ways, the fourth blank 26 ways, the
fifth blank 10 ways, the sixth blank 10 ways, and the seventh
blank 10 ways. By the Multiplication Principle, there are
26×26×26×26×10×10×10 = 456,976,000 possible
license plates.
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Slide 9- 9
Permutations of an n-Set
There are n! permutations of an n-set.
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Slide 9- 10
Example Distinguishable Permutations
Count the number of different 8-letter “words” that can be
formed using the letters in the word COMPUTER.
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Slide 9- 11
Example Distinguishable Permutations
Count the number of different 8-letter “words” that can be
formed using the letters in the word COMPUTER.
Each permutation of the 8 letters forms a different word.
There are 8! = 40,320 such permutations.
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Slide 9- 12
Distinguishable Permutations
There are n ! distinguishable permutations of an n-set containing n
distinguishable objects.
If an n-set contains n objects of a first kind, n objects of a second
1
2
kind, and so on, with n  n  ...  n  n, then the number of
1
2
k
distinguishable permutations of the n-set is
n!
.
n !n !n ! n !
1
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2
3
k
Slide 9- 13
Permutations Counting Formula
The number of permutations of n objects taken r at a time is
n!
denoted P and is given by P 
for 0  r  n.
 n  r !
n
r
n
r
If r  n, then P  0.
n
r
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Slide 9- 14
Combination Counting Formula
The number of combinations of n objects taken r at a time is
n!
denoted C and is given by C 
for 0  r  n.
r ! n  r !
n
r
n
r
If r  n, then C  0.
n
r
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Slide 9- 15
Example Counting Combinations
How many 10 person committees can be formed from a group of
20 people?
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Slide 9- 16
Example Counting Combinations
How many 10 person committees can be formed from a group of
20 people?
Notice that order is not important. Using combinations,
20!
C 
 184, 756.
10! 20  10 !
20
10
There are 184,756 possible committees.
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Slide 9- 17
Formula for Counting Subsets of an n-Set
n
There are 2 subsets of a set with n objects (including the
empty set and the entire set).
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Slide 9- 18
9.2
The Binomial Theorem
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Quick Review
Use the distributive property to expand the binomial.
1.  x  y 
2
2. (a  2b)
3. (2c  3d )
4. (2 x  y )
2
2
2
5.  x  y 
3
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Slide 9- 20
Quick Review Solutions
Use the distributive property to expand the binomial.
1.  x  y 
x  2 xy  y
2
2
2
2. (a  2b) a  4ab  4b
3. (2c  3d ) 4c  12cd  9d
4. (2 x  y ) 4 x  4 xy  y
2
2
2
2
2
2
5.  x  y 
3
2
2
2
x  3x y  3 xy  y
3
2
2
3
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Slide 9- 21
What you’ll learn about




Powers of Binomials
Pascal’s Triangle
The Binomial Theorem
Factorial Identities
… and why
The Binomial Theorem is a marvelous study in
combinatorial patterns.
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Slide 9- 22
Binomial Coefficient
The binomial coefficients that appear in the expansion of (a  b)
n
are the values of C for r  0,1, 2,3,..., n.
n
r
A classical notation for C , especially in the context of binomial
n
r
n
coefficients, is   . Both notations are read "n choose r."
r
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Slide 9- 23
Example Using nCr to Expand a Binomial
Expand  a  b  , using a calculator to compute the binomial coefficients.
4
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Slide 9- 24
Example Using nCr to Expand a Binomial
Expand  a  b  , using a calculator to compute the binomial coefficients.
4
0,1, 2,3, 4 into the calculator to find the binomial
coefficients for n  4. The calculator returns the list 1,4,6,4,1 .
Enter 4 C
n
r
Using these coefficients, construct the expansion:
a  b
4
 a  4a b  6a b  4ab  b .
4
3
2
2
3
4
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Slide 9- 25
Recursion Formula for Pascal’s Triangle
 n   n  1  n  1
 r    r  1    r  or, equivalently, C  C  C
  
 

n
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r
n 1
r 1
n 1
r
Slide 9- 26
The Binomial Theorem
For any positive integer n,
n
n
n
 n
 a  b     a    a b  ...    a b  ...    b ,
0
1
r
 n
n
n!
where    C 
.
r !(n  r )!
r
n
n
n
n 1
nr
r
n
r
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Slide 9- 27
Basic Factorial Identities
For any integer n  1, n !  n  n  1!
For any integer n  0,  n  1!   n  1 n !
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Slide 9- 28
9.3
Probability
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Quick Review
How many outcomes are possible for the following experiments.
1. Two coins are tossed.
2. Two different 6-sided dice are rolled.
3. Two chips are drawn simultaneously without replacement from
a jar with 8 chips.
4. Two different cards are drawn from a standard deck of 52.
C
5. Evaluate without using a calculator.
C
4
2
8
2
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Slide 9- 30
Quick Review Solutions
How many outcomes are possible for the following experiments.
1. Two coins are tossed. 4
2. Two different 6-sided dice are rolled. 36
3. Two chips are drawn simultaneously without replacement from
a jar with 8 chips. 28
4. Two different cards are drawn from a standard deck of 52. 1326
C
5. Evaluate without using a calculator.
3/14
C
4
2
8
2
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Slide 9- 31
What you’ll learn about





Sample Spaces and Probability Functions
Determining Probabilities
Venn Diagrams and Tree Diagrams
Conditional Probability
Binomial Distributions
… and why
Everyone should know how mathematical the “laws of
chance” really are.
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Slide 9- 32
Probability of an Event (Equally Likely
Outcomes)
If E is an event in a finite, nonempty sample space S of equally likely
outcomes, then the probability of the event E is
the number of outcomes in E
P( E ) 
.
the number of outcomes in S
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Slide 9- 33
Probability Distribution for the Sum of Two
Fair Dice
Outcome
2
3
4
5
6
7
8
9
10
11
12
Probability
1/36
2/36
3/36
4/36
5/36
6/36
5/36
4/36
3/36
2/36
1/36
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Slide 9- 34
Example Rolling the Dice
Find the probability of rolling a sum divisible by 4 on a single
roll of two fair dice.
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Slide 9- 35
Example Rolling the Dice
Find the probability of rolling a sum divisible by 4 on a single
roll of two fair dice.
The event E consists of the outcomes 4,8,12 . To get the probability
of E we add up the probabilities of the outcomes in E:
3
5
1
9 1
P( E ) 
 

 .
36 36 36 36 4
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Slide 9- 36
Probability Function
A probability function is a function P that assigns a real number
to each outcome in a sample space S subject to the following conditions:
1. 0  P(O)  1;
2. the sum of the probabilities of all outcomes in S is 1;
3. P()  0.
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Slide 9- 37
Probability of an Event (Outcomes not
Equally Likely)
Let S be a finite, nonempty sample space in which every outcome
has a probability assigned to it by a probability function P. If E is
any event in S , the probability of the event E is the sum of the
probabilities of all the outcomes contained in E.
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Slide 9- 38
Strategy for Determining Probabilities
1. Determine the sample space of all possible outcomes. When possible,
choose outcomes that are equally likely.
2. If the sample space has equally likely outcomes, the probability of an
the number of outcomes in E
event E is determined by counting: P ( E ) 
.
the number of outcomes in S
3. If the sample space does not have equally likely outcomes, determine
the probability function. (This is not always easy to do.) Check to be sure
that the conditions of a probability function are satisfied. Then the
probability of an event E is determined by adding up the probabilities
of all the outcomes contained in E.
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Slide 9- 39
Example Choosing Chocolates
Dylan opens a box of a dozen chocolate cremes and offers three of them to
Russell. Russell likes vanilla cremes the best, but all the chocolates look alike
on the outside. If five of the twelve cremes are vanilla, what is the probability
that all of Russell’s picks are vanilla?
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Slide 9- 40
Example Choosing Chocolates
Dylan opens a box of a dozen chocolate cremes and offers three of them to
Russell. Russell likes vanilla cremes the best, but all the chocolates look alike
on the outside. If five of the twelve cremes are vanilla, what is the probability
that all of Russell’s picks are vanilla?
The experiment in question is the selection of three chocolates,
without regard to order, from a box of 12. There are C  220
12
3
outcomes of this experiment. The event E consists of all possible
combinations of 3 that can be chosen, without regard to order, from
the 5 vanilla cremes available. There are C  10 ways.
5
3
Therefore, P( E )  10 / 220  1/ 22.
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Slide 9- 41
Multiplication Principle of Probability
Suppose an event A has probability p1 and an
event B has probability p2 under the
assumption that A occurs. Then the probability
that both A and B occur is p1p2.
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Slide 9- 42
Example Choosing Chocolates
Dylan opens a box of a dozen chocolate cremes and offers three of them to
Russell. Russell likes vanilla cremes the best, but all the chocolates look alike
on the outside. If five of the twelve cremes are vanilla, what is the probability
that all of Russell’s picks are vanilla?
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 9- 43
Example Choosing Chocolates
Dylan opens a box of a dozen chocolate cremes and offers three of them to
Russell. Russell likes vanilla cremes the best, but all the chocolates look alike
on the outside. If five of the twelve cremes are vanilla, what is the probability
that all of Russell’s picks are vanilla?
The probability of picking a vanilla creme on the first draw is 5/12.
Under the assumption that a vanilla creme was selected in the first
draw, the probability of picking a vanilla creme on the second draw is
4/11. Under the assumption that a vanilla creme was selected in the first
and second draw, the probability of picking a vanilla creme on the third
draw is 3/10. By the Multiplication Principle, the probability of picking
5 4 3
60
1
a vanilla creme on all three picks is
  
 .
12 11 10 1320 22
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Slide 9- 44
Conditional Probability Formula
If the event B depends on the event A, then P( B | A) 
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P( A and B)
.
P( A)
Slide 9- 45
Binomial Distribution
Suppose an experiment consists of n-independent repetitions of an
experiment with two outcomes, called "success" and "failure." Let
P(success)  p and P(failure)  q. (Note that q  1  p.)
Then the terms in the binomial expansion of ( p  q) give the respective
probabilities of exactly n, n  1,..., 2, 1, 0 successes.
Number of successes out of
Probability
n independent repetitions
n
p
n
n
n 1
1
0
 n 
 n  1 p q


n 1
n
 r  pq
 
q
n 1
n
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Slide 9- 46
Example Shooting Free Throws
Suppose Tommy makes 92% of his free throws. If he shoots 15 free throws,
and if his chance of making each one is independent of the other shots, what is
the probability that he makes all 15?
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Slide 9- 47
Example Shooting Free Throws
Suppose Tommy makes 92% of his free throws. If he shoots 15 free throws,
and if his chance of making each one is independent of the other shots, what is
the probability that he makes all 15?
P(15 successes)   0.92   0.286
15
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Slide 9- 48
Example Shooting Free Throws
Suppose Tommy makes 92% of his free throws. If he shoots 15 free throws,
and if his chance of making each one is independent of the other shots, what is
the probability that he makes exactly 10?
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Slide 9- 49
Example Shooting Free Throws
Suppose Tommy makes 92% of his free throws. If he shoots 15 free throws,
and if his chance of making each one is independent of the other shots, what is
the probability that he makes exactly 10?
 15 
P(10 successes)=    0.92   0.08   0.00427
10 
10
5
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Slide 9- 50
9.4
Sequences
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Quick Review
Evaluate each expression when a  3, r  2, n  4 and d  2.
1. a  (n  1)d
2. a  r
Find a .
n 1
10
k 1
k
4. a  2  3
3. a 
k
k 1
k
5. a  a  3 and a  10
k
k 1
9
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Slide 9- 52
Quick Review Solutions
Evaluate each expression when a  3, r  2, n  4 and d  2.
1. a  (n  1)d 9
2. a  r
24
Find a .
n 1
10
k 1
k
4. a  2  3
3. a 
k
k 1
k
11
10
39,366
5. a  a  3 and a  10
k
k 1
9
13
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Slide 9- 53
What you’ll learn about




Infinite Sequences
Limits of Infinite Sequences
Arithmetic and Geometric Sequences
Sequences and Graphing Calculators
… and why
Infinite sequences, especially those with finite limits,
are involved in some key concepts of calculus.
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Slide 9- 54
Limit of a Sequence
Let a  be a sequence of real numbers, and consider lim a .
n
n 
n
If the limit is a finite number L, the sequence converges and L
is the limit of the sequence. If the limit is infinite or nonexistent,
the sequence diverges.
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Slide 9- 55
Example Finding Limits of Sequences
Determine whether the sequence converges or diverges. If it converges,
give the limit.
2 1 2
2
2,1, , , ,..., ,...
3 2 5
n
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Slide 9- 56
Example Finding Limits of Sequences
Determine whether the sequence converges or diverges. If it converges,
give the limit.
2 1 2
2
2,1, , , ,..., ,...
3 2 5
n
lim
n
2
 0, so the sequence converges to a limit of 0.
n
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Slide 9- 57
Arithmetic Sequence
A sequence a  is an arithmetic sequence if it can be written in the
n
form a, a  d , a  2d ,..., a  ( n  1) d ,... for some constant d .
The number d is called the common difference.
Each term in an arithmetic sequence can be obtained recursively from
its preceding term by adding d : a  a  d (for all n  2).
n
n 1
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Slide 9- 58
Example Arithmetic Sequences
Find (a) the common difference, (b) the tenth term, (c) a recursive rule for the
nth term, and (d) an explicit rule for the nth term.
-2, 1, 4, 7, …
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Slide 9- 59
Example Arithmetic Sequences
Find (a) the common difference, (b) the tenth term, (c) a recursive rule for the
nth term, and (d) an explicit rule for the nth term.
-2, 1, 4, 7, …
(a) The common difference is 3.
(b) a  2  (10  1)3  25
10
(c) a  2 a  a  3 for all n  2
1
n
n 1
(d) a  2  3(n  1)  3n  5
n
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Slide 9- 60
Geometric Sequence
A sequence a  is a geometric sequence if it can be written in the
n
form a, a  r , a  r ,..., a  r ,... for some nonzero constant r.
2
n 1
The number r is called the common ratio.
Each term in a geometric sequence can be obtained recursively from
its preceding term by multiplying by r : a  a  r (for all n  2).
n
n 1
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Slide 9- 61
Example Defining Geometric Sequences
Find (a) the common ratio, (b) the tenth term, (c) a recursive rule for the
nth term, and (d) an explicit rule for the nth term.
2, 6, 18,…
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Slide 9- 62
Example Defining Geometric Sequences
Find (a) the common ratio, (b) the tenth term, (c) a recursive rule for the
nth term, and (d) an explicit rule for the nth term.
2, 6, 18,…
(a) The ratio is 3.
(b) a  2  3  39,366
10 1
10
(c) a  2 and a  3a
1
n
n 1
for n  2.
(d) a  2  3 .
n 1
n
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Slide 9- 63
Sequences and Graphing Calculators


One way to graph a explicitly defined sequences is as scatter
plots of the points of the form (k,ak).
A second way is to use the sequence mode on a graphing
calculator.
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Slide 9- 64
The Fibonacci Sequence
The Fibonacci sequences can be defined recursively by
a 1
1
a 1
2
a  a a
n
n2
n 1
for all positive integers n  3.
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Slide 9- 65
9.5
Series
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Quick Review
a  is an arithmetic sequence. Use the given information to find a
n
10
.
1. a  5; d  4
1
2. a  5; d  2
3
a  is a geometric sequence. Use the given information to find a
n
10
.
3. a  5; r  4
1
4. a  5; r  4
3
5. Find the sum of the first 3 terms of the sequence n  .
2
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Slide 9- 67
Quick Review Solutions
a  is an arithmetic sequence. Use the given information to find a
n
10
1. a  5; d  4
41
2. a  5; d  2
19
1
3
a  is a geometric sequence. Use the given information to find a
n
10
.
.
3. a  5; r  4 1,310,720
1
4. a  5; r  4
3
81,920
5. Find the sum of the first 3 terms of the sequence n  . 14
2
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 9- 68
What you’ll learn about




Summation Notation
Sums of Arithmetic and Geometric Sequences
Infinite Series
Convergences of Geometric Series
… and why
Infinite series are at the heart of integral calculus.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 9- 69
Summation Notation
In summation notation, the sum of the terms of the sequence a , a ,..., a 
1
2
n
is denoted  a which is read "the sum of a from k  1 to n."
n
k 1
k
k
The variable k is called the index of summation.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 9- 70
Sum of a Finite Arithmetic Sequence
Let a , a ,..., a  be a finite arithmetic sequence with common difference d .
1
2
n
Then the sum of the terms of the sequence is
 a  a  a  ...  a
n
k 1
k
1
2
n
a a 
 n

 2 
n
  2a  (n  1)d 
2
1
n
1
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 9- 71
Example Summing the Terms of an
Arithmetic Sequence
A corner section of a stadium has 6 seats along the front row. Each
successive row has 3 more seats than the row preceding it. If the top
row has 24 seats, how many seats are in the entire section?
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 9- 72
Example Summing the Terms of an
Arithmetic Sequence
A corner section of a stadium has 6 seats along the front row. Each
successive row has 3 more seats than the row preceding it. If the top
row has 24 seats, how many seats are in the entire section?
The number of seats in the rows form an arithmetic sequence with
a  6, a  24, and d  3. Solving
1
n
a  a  (n  1)d
n
1
24  6  3(n  1)
n7
Apply the Sum of a Finite Sequence Theorem:
 6  24 
Sum of chairs  7 
  105. There are 105 seats in the section.
 2 
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Slide 9- 73
Sum of a Finite Geometric Sequence
Let a , a ,..., a  be a finite geometric sequence with common ratio r.
1
2
n
Then the sum of the terms of the sequence is
 a  a  a  ...  a
n
k 1
k
1

2
a 1  r
1
n
n

1 r
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 9- 74
Infinite Series

An infinite series is an expression of the form  a  a  a  ...  a  ...
k 1
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
k
1
2
n
Slide 9- 75
Sum of an Infinite Geometric Series

The geometric series  a  r
k 1
k 1
converges if and only if | r | 1.
a
If it does converge, the sum is
.
1 r
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 9- 76
Example Summing Infinite Geometric
Series
Determine whether the series converges. If it converges, give the sum.
 2  0.25 

k 1
k 1
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 9- 77
Example Summing Infinite Geometric
Series
Determine whether the series converges. If it converges, give the sum.
 2  0.25 

k 1
k 1
Since |r | 0.25  1, the series converges.
a
2
8
The sum is

 .
1  r 1  0.25 3
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 9- 78
9.6
Mathematical Induction
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Quick Review
1. Expand the product k ( k  2)( k  4).
Factor the polynomial.
2. n  7 n  10
3. n  3n  3n  1
2
3
2
x
.
x 1
5. Find f (t  1) given f ( x)  x  1.
4. Find f (t ) given f ( x) 
2
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 9- 80
Quick Review Solutions
1. Expand the product k (k  2)(k  4). k  6k  8k
3
2
Factor the polynomial.
 n  2  n  5
 3n  1  n  1
2. n  7 n  10
2
3. n  3n
3
2
3
x
t
4. Find f (t ) given f ( x) 
.
x 1 t 1
5. Find f (t  1) given f ( x)  x  1. t  2t  2
2
2
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 9- 81
What you’ll learn about



The Tower of Hanoi Problem
Principle of Mathematical Induction
Induction and Deduction
… and why
The principle of mathematical induction is a
valuable technique for proving combinatorial formulas.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 9- 82
The Tower of Hanoi Solution
The minimum number of moves required to
move a stack of n washers in a Tower of Hanoi
game is 2n – 1.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 9- 83
Principle of Mathematical Induction
Let Pn be a statement about the integer n. Then
Pn is true for all positive integers n provided the
following conditions are satisfied:
1. (the anchor) P1 is true;
2. (inductive step) if Pk is true, then Pk+1 is
true.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 9- 84
9.7
Statistics and Data (Graphical)
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Quick Review
Solve for the require value.
1. 567 is what percent of 12345?
2. 73 is what percent of 360 ?
3. 357 is 35.7% of what number?
Round the given value to the nearest whole number
in the specified units.
4. 1234 millions (billions)
5. 1,234,567 (millions)
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 9- 86
Quick Review Solutions
Solve for the require value.
1. 567 is what percent of 12345?  4.593%
2. 73 is what percent of 360 ?  20.278%
3. 357 is 35.7% of what number? 1000
Round the given value to the nearest whole number
in the specified units.
4. 1234 millions (billions) 1 billion
5. 1,234,567 (millions) 1 million
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 9- 87
What you’ll learn about






Statistics
Displaying Categorical Data
Stemplots
Frequency Tables
Histograms
Time Plots
… and why
Graphical displays of data are increasingly prevalent in
professional and popular media. We all need to understand them.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 9- 88
Leading Causes of Death in the United
States in 2001
Cause of Death
Number of Deaths
Percentage
Heart Disease
Cancer
Stroke
Other
700,142
553,768
163,538
1,018,977
29.0
22.9
6.8
41.3
Source: National Center for Health Statistics, as reported in The World Almanac
and Book of Facts 2005.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 9- 89
Bar Chart, Pie Chart, Circle Graph
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 9- 90
Example Making a Stemplot
Make a stemplot for the
given data.
12.3
23.4
12.0
24.5
23.7
18.7
22.4
19.5
24.5
24.6
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 9- 91
Example Making a Stemplot
Make a stemplot for the
given data.
12.3
23.4
12.0
24.5
23.7
18.7
22.4
19.5
24.5
24.6
Stem Leaf
12
0,3
18
7
19
5
22
4
23
4,7
24
5,5,6
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 9- 92
Time Plot
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Slide 9- 93
9.8
Statistics and Data (Algebraic)
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Quick Review
Write the sum in expanded form.
5
1.  x
i
i 0
1
2.  x
3
1
3.  x  x
3
Write the sum in sigma notation
4. x f  x f  x f  x f
3
i
i 1
3
i 1
2
2
1
5.
20


i
3
3
4
4
5
5
 x  x    x  x    x  x    x  x  
2
1
2
2
2
3
2
4
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 9- 95
Quick Review Solutions
Write the sum in expanded form.
x x  x x  x x
5
1.  x
i 0
i
0
1
2
3
4
5
1
1
2.  x
x  x  x 
3
3
1
1
3.  x  x
x x x xx x
3
3
Write the sum in sigma notation
3
i
i 1
3
i 1

1
2
 
i
3
1
2
4. x f  x f  x f  x f
2
2
1
5.
20
3
3
4
4
5
3

5
5
x f
i
i 2
i
 x  x    x  x    x  x    x  x  
2
1
2
2
2
3
2
4
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

1
 x x
20
4
i 1
i

2
Slide 9- 96
What you’ll learn about






Parameters and Statistics
Mean, Median, and Mode
The Five-Number Summary
Boxplots
Variance and Standard Deviation
Normal Distributions
… and why
The language of statistics is becoming more commonplace in our
everyday world.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 9- 97
Mean
The mean of a list of n numbers  x , x ,..., x  is
1
2
n
x  x  ...  x
1
x
 x.
n
n
1
2
n
n
i 1
i
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 9- 98
Median
The median of a list of n numbers {x1,x2,…,xn}
arranged in order (either ascending or
descending) is
 the middle number if n is odd, and
 the mean of the two middle numbers if n is
even.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 9- 99
Mode
The mode of a list of numbers is the number that
appears most frequently in the list.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 9- 100
Example Finding Mean, Median, and
Mode
Find the (a) mean, (b) median, and (c) mode of the data:
3, 6, 5, 7, 8, 10, 6, 2, 4, 6
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 9- 101
Example Finding Mean, Median, and
Mode
Find the (a) mean, (b) median, and (c) mode of the data:
3, 6, 5, 7, 8, 10, 6, 2, 4, 6
3  6  5  7  8  10  6  2  4  6
(a) x 
 5.7
10
(b) Put the data in order: 2, 3, 4, 5, 6, 6, 6, 7, 8, 10.
The median is 6.
(c) The mode is 6.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 9- 102
Weighted Mean
The formula for finding the mean of a list of numbers  x , x ,..., x  with
1
2
n
n
frequencies  f , f ,..., f
1
2
n

x f
x f  x f  ...  x f
is x 

.
f  f  ...  f
f
1
1
2
2
n
n
i 1
i
i
n
1
2
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
n
i 1
i
Slide 9- 103
Five-Number Summary
The five - number summary of a data set is the collection
minimum, Q , median, Q , maximum .
1
3
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Slide 9- 104
Boxplot
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 9- 105
Outlier
A number in a data set can be considered an
outlier if it is more than 1.5×IQR below the
first
quartile or above the third quartile.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 9- 106
Standard Deviation
The standard deviation of the numbers  x , x ,..., x  is
1


2
n
1

 x  x , where x denotes the mean.
n
The variance is  , the square of the standard deviation.
n
i 1
i
2
2
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 9- 107
Normal Curve
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Slide 9- 108
The 68-95-99.7 Rule
If the data for a population are normally distributed with
mean μ and standard deviation σ, then
 Approximately 68% of the data lie between μ - 1σ
and μ + 1σ.
 Approximately 95% of the data lie between μ - 2σ
and μ + 2σ.
 Approximately 99.7% of the data lie between μ - 3σ
and μ + 3σ.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 9- 109
The 68-95-99.7 Rule
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 9- 110
Chapter Test
1. A travel agent is trying to schedule a client's trip from city A
to city B. There are three direct flights, three flights from A to
a connecting city C, and four flights from this connecting city
C to city B. How many trips are possible?
2. A club has 45 members, and its membership committee has
three members. How many different membership committees
are possible?
3. Find the coefficient of x y in the expansion of  2 x  y  .
2
6
8
4. List the elements in the sample space. A game spinner on a
circular region divided into 6 equal sectors numbered 1 - 6 is spun.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 9- 111
Chapter Test
5. A fair coin is tossed four times. Find the probability of obtaining
one head and three tails.
6. Two cans of mixed nuts of different brands are open on a table.
Brand A consists of 30% cashews, while brand B consists of 40%
cashews. A can is chosen at random, and a nut is chosen at random
from the can. Find the probability that the nut is
(a) from the brand A can.
(b) a brand A cashew.
(c) a cashew.
(d) from the brand A can, given that it is a cashew.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 9- 112
Chapter Test
7. Find the first 6 terms and the 12th term of the sequence given
b  5 and b  2b , for k  2.
1
k
k 1
8.Find an explicit formula for the nth term of the arithmetic sequence
5, 1,3, 7,...
1 1
1
9. Find the sum of the terms of the geometric sequence  3, 1,  ,  , 
3 9 27
10. Determine whether the geometric series  3  0.5  converges. If it does,

k
k 1
find its sum.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 9- 113
Chapter Test Solutions
1. A travel agent is trying to schedule a client's trip from city A
to city B. There are three direct flights, three flights from A to
a connecting city C, and four flights from this connecting city
C to city B. How many trips are possible? 15
2. A club has 45 members, and its membership committee has
three members. How many different membership committees
are possible? 14,190
3. Find the coefficient of x y in the expansion of  2 x  y  .112
2
6
8
4. List the elements in the sample space. A game spinner on a
circular region divided into 6 equal sectors numbered 1 - 6 is spun.
1,2,3,4,5,6
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 9- 114
Chapter Test Solutions
5. A fair coin is tossed four times. Find the probability of obtaining
one head and three tails. 1/4
6. Two cans of mixed nuts of different brands are open on a table.
Brand A consists of 30% cashews, while brand B consists of 40%
cashews. A can is chosen at random, and a nut is chosen at random
from the can. Find the probability that the nut is
(a) from the brand A can. 0.5
(b) a brand A cashew. 0.15
(c) a cashew. 0.35
(d) from the brand A can, given that it is a cashew.  0.43
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 9- 115
Chapter Test Solutions
7. Find the first 6 terms and the 12th term of the sequence given
b  5 and b  2b , for k  2. 5,10, 20, 40, 80,160; 10,240
1
k 1
k
8. Find an explicit formula for the nth term of the arithmetic sequence
5, 1,3, 7,... a  4n  9
n
1 1
1
9. Find the sum of the terms of the geometric sequence  3, 1,  ,  , 
3 9 27
121/ 27
10. Determine whether the geometric series  3  0.5  converges. If it does,

k
k 1
find its sum. converges; 3
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Slide 9- 116