Download Chapter 9

Barnett/Ziegler/Byleen
Precalculus: A Graphing Approach
Chapter Nine
Sequences, Series, and Probability
Copyright © 2000 by the McGraw-Hill Companies, Inc.
Sequences and Series
Compact Notation
A finite sequence
1, 3, 5, 7, 9
an = 2n - 1,
n = 1, 2, ... ,5
The corresponding finite series
5
1+3+5+7+9
 (2k  1)
k 1
A finite sequence
-1 1 -1 1 -1
1, 2 , 3 , 4 , 5 , 6
(-1)n+1
an =
, n = 1,2, ... ,6
n
The corresponding finite alternating series
1
1
1
1
1
1 - 2 + 3 - 4 + 5 -6
(1)k  1

k
k 1
6
Copyright © 2000 by the McGraw-Hill Companies, Inc.
9-1-95(a)
Sequences and Series
Compact Notation
An infinite sequence
1 1 1
1
,
,
,
, ...
2 4 8 16
an =
1
,
n
2
n = 1, 2, 3, . . .
The corresponding infinite series
1
1
1
1
+
+
+
2
4
8
16

+ ...
1
k
k 1 2

The compact form for representing a series makes use of the
summation sign

and the summing index k.
Copyright © 2000 by the McGraw-Hill Companies, Inc.
9-1-95(b)
Principle of Mathematical Induction
Let Pn be a statement associated with
each positive integer n, and suppose the
following conditions are satisfied:
1. P1 is true.
2. For any positive integer k, if Pk is true,
then Pk + 1 is also true.
Then the statement Pn is true for all positive
integers n.
+
Condition
domino
(a) Condition1:1:The
The first
first domino
pushedover.
over.
cancan
be be
pushed
Condition
thekth
kth
domino
(b)
Condition2:
2: If
If the
domino
falls,
doesthe
the(k+1)st.
(k+1)st.
falls,then
then so
so does
Conclusion:
thedominos
dominos
(c)
Conclusion:All
All the
willwill
fall.
fall.
Copyright © 2000 by the McGraw-Hill Companies, Inc.
9-2-96
Arithmetic Sequences and Series
A sequence
a ,a ,a ,…,a ,…
1 2 3
n
is called an arithmetic sequence, or
arithmetic progression, if there exists a constant d, called the common
difference, such that
a =a
+d
n
n–1
for every n > 1.
If a , a , a , … is an arithmetic sequence, then the corresponding
1 2 3
series a + a + a + … is an arithmetic series.
1
2
3
The nth Term of an Arithmetic Sequence
a = a + (n – 1)d
n
1
Sum of the First n Terms of an Arithmetic Series
First Form:
Second Form:
n
S = 2 [2a + (n – 1)d]
n
1
n
S = 2 (a + a )
n
1
n
Copyright © 2000 by the McGraw-Hill Companies, Inc.
9-3-97
Geometric Sequences and Series
A sequence
a , a , a , … , a , … is called a geometric sequence, or
1 2 3
n
geometric progression, if there exists a nonzero constant r, called the
common ratio, such that
a = ra
n
n–1
for every n > 1.
If a , a , a , … is a geometric sequence, then the corresponding series
1 2 3
a + a + a + … is a geometric series.
1
2
3
The nth Term of a Geometric Sequence
a =a r
n
n–1
1
Sum of the First n Terms of a Geometric Series
a -a r
First Form:
Second Form:
S =
n
1
1
1–r
a - ra
1
S = 1–r
n
n
n
r1
r1
Sum of an Infinite Geometric Series
a
1
S =1–r
|r| < 1

Copyright © 2000 by the McGraw-Hill Companies, Inc.
9-3-98
Multiplication Principle
Coin
Outcomes
Die
Outcomes
1
2
H
3
4
5
6
Start
1
2
H
T
Heads
Tails
T
3
4
5
6
Coin Outcomes
Combined
Outcomes
(H,
(H,
(H,
(H,
(H,
(H,
1)
2)
3)
4)
5)
6)
(T,
(T,
(T,
(T,
(T,
(T,
1)
2)
3)
4)
5)
6)
Die Outcomes
Copyright © 2000 by the McGraw-Hill Companies, Inc.
9-4-99
Permutations and Combinations
The number of permutations of n objects taken r at a time
is given by
= n (n - 1) (n - 2)  . . .  (n - r + 1)
Pn,r
or
Pn,r
n!
= (n - r)!
0  r  n
The number of combinations of n objects taken r at a
time is given by
Cn,r
=
n

r





Pn,r
= r!
n!
= r!(n - r)!
Copyright © 2000 by the McGraw-Hill Companies, Inc.
0  r  n
9-4-100
Outcomes of Rolling Two Dice
•
•
••
• • • • • •• ••
• • • • • • •
•
(1, 1) (1, 2) (1, 3) (1, 4) (1, 5) (1, 6)
•
••
•
• •
• •
•••
• •
•• ••
• •
(2, 1) (2, 2) (2, 3) (2, 4) (2, 5) (2, 6)
•
First Die
•
Second Die
(3, 1) (3, 2) (3, 3) (3, 4) (3, 5) (3, 6)
(4, 1) (4, 2) (4, 3) (4, 4) (4, 5) (4, 6)
(5, 1) (5, 2) (5, 3) (5, 4) (5, 5) (5, 6)
(6, 1) (6, 2) (6, 3) (6, 4) (6, 5) (6, 6)
Copyright © 2000 by the McGraw-Hill Companies, Inc.
9-5-101
Probabilities for Simple Events
Given a sample space
S = {e1, e2, … , en}
with n simple events, to each simple event ei we assign a real number
denoted by P(ei ), called the probability of the event ei. These numbers
can be assigned in an arbitrary manner as long as the following two
conditions are satisfied:
1. The probability of a simple event is a number between 0 and 1,
inclusive. That is,
0  P(ei)  1
2. The sum of the probabilities of all simple events in the sample
space is 1. That is,
P(ei) + P(e2) + … + P(en) = 1
Any probability assignment that meets conditions 1 and 2 is said to be
an acceptable probability assignment.
Copyright © 2000 by the McGraw-Hill Companies, Inc.
9-5-102
Probability of an Event E
Given an acceptable probability assignment for the simple events in
a sample space S, we define the probability of an arbitrary event E,
denoted by P(E), as follows:
1. If E is the empty set, then P(E) = 0.
2. If E is a simple event, then P(E) has already been assigned.
3. If E is a compound event, then P(E) is the sum of the probabilities
of all the simple events in E.
4. If E is the sample space S, then P(E) = P(S) = 1 [this is a special
case of 3].
Copyright © 2000 by the McGraw-Hill Companies, Inc.
9-5-103
Equally Likely Assumption
Probability of a Simple Event
If, in a sample space
S = {e1, e2, … , en}
with n elements, we assume each simple event ei is as likely to occur
1
as any other, then we assign the probability to each. That is,
n
1
P(ei ) 
n
Probability of an Arbitrary Event
If we assume each simple event in sample space S is as likely to occur as
any other, then the probability of an arbitrary event E in S is given by
P(E) 
Number of elements in E n(E)

Number of elements in S n(S)
Copyright © 2000 by the McGraw-Hill Companies, Inc.
9-5-104
Binomial Formula
(a + b)1 = a + b
(a + b)2 = a2 + 2ab + b2
(a + b)3 = a3 + 3a2 b + 3ab2 + b3
(a + b)4 = a 4 + 4a3 b + 6a2 b2 + 4ab 3 + b4
(a + b)5 = a5 + 5a 4 b + 10a3 b2 + 10a2 b3 + 5ab 4 + b5
In general:
n
(a + b)
n n n n – 1
n n – 2 2 n n – 3 3
n n
 
 
 
 
 
=  a +  a
b+  a
b +  a
b +…+  b
n
0
1
2
3
 n
n  
n–k k
=  a
b
k 0  
k
Copyright © 2000 by the McGraw-Hill Companies, Inc.
9-6-105