C. The General Principle of Mathematical Induction
Tải bản đầy đủ - 0trang
cob19545_ch09_797-805.qxd
12/17/10
12:54 AM
Page 800
College Algebra Graphs & Models—
800
9–40
CHAPTER 9 Additional Topics in Algebra
2kϩ1 ϭ 212k 2
Ն 21k ؉ 12
Ն 2k ϩ 2
properties of exponents
induction hypothesis: substitute k ϩ 1 for 2k
(symbol changes since k ϩ 1 is less than or equal to 2k)
distribute
Since k is a positive integer, 2kϩ1 Ն 2k ϩ 2 Ն k ϩ 2,
showing 2kϩ1 Ն k ϩ 2.
WORTHY OF NOTE
Note there is no reference to an, ak,
or ak+1 in the statement of the
general principle of mathematical
induction.
EXAMPLE 4
Since the truth of Pkϩ1 follows from Pk, the formula is true for all n.
Now try Exercises 39 through 42
ᮣ
ᮣ
Proving Divisibility Using Mathematical Induction
Let Pn be the statement, “4n Ϫ 1 is divisible by 3 for all positive integers n.” Use
mathematical induction to prove that Pn is true.
Solution
ᮣ
If a number is evenly divisible by three, it can be written as the product of 3 and
some positive integer we will call p.
1. Show Pn is true for n ϭ 1:
Pn: 4n Ϫ 1 ϭ 3p
P1: 4112 Ϫ 1 ϭ 3p
3 ϭ 3p ✓
given statement, p ʦ ޚ
substitute 1 for n
statement is true for n ϭ 1
2. Assume that Pk is true.
Pk:
4k Ϫ 1 ϭ 3p
4k ϭ 3p ϩ 1
induction hypothesis
isolate 4k
and use it to show the truth of Pkϩ1. That is,
Pkϩ1:
4kϩ1 Ϫ 1 ϭ 3q for q ʦ ޚis also true.
Beginning with the left-hand side we have:
4kϩ1 Ϫ 1 ϭ 4 # 4k Ϫ 1
ϭ 4 # 13p ؉ 12 Ϫ 1
ϭ 12p ϩ 3
ϭ 314p ϩ 12 ϭ 3q
properties of exponents
induction hypothesis: substitute 3p ϩ 1 for 4k
distribute and simplify
factor
The last step shows 4
Ϫ 1 is divisible by 3. Since the original statement is
true for n ϭ 1, and the truth of Pk implies the truth of Pkϩ1, the statement,
“4n Ϫ 1 is divisible by 3” is true for all positive integers n.
kϩ1
Now try Exercises 43 through 47
C. You’ve just seen how
we can apply the principle of
mathematical induction to
general statements involving
natural numbers
ᮣ
We close this section with some final notes. Although the base step of a proof by
induction seems trivial, both the base step and the induction hypothesis are necessary
1
1
parts of the proof. For example, the statement n 6
is false for n ϭ 1, but true for
3
3n
all other positive integers. Finally, for a fixed natural number p, some statements are
false for all n 6 p, but true for all n Ն p. By modifying the base case to begin at p, we
can use the induction hypothesis to prove the statement is true for all n greater than p.
For example, n 6 13n2 is false for n 6 4, but true for all n Ն 4.
cob19545_ch09_797-805.qxd
11/10/10
7:35 PM
Page 801
College Algebra Graphs & Models—
9–41
801
Section 9.4 Mathematical Induction
9.4 EXERCISES
ᮣ
CONCEPTS AND VOCABULARY
Fill in the blank with the appropriate word or phrase. Carefully reread the section if needed.
1. No
statement
number of verifications can prove a
true.
3. Assuming that a statement/formula is true for
n ϭ k is called the
.
5. Explain the equation Sk ϩ akϩ1 ϭ Skϩ1. Begin by
saying, “Since the kth term is arbitrary p ” (continue
from here).
ᮣ
7. an ϭ 10n Ϫ 6
9. an ϭ n
11. an ϭ 2nϪ1
8. an ϭ 6n Ϫ 4
10. an ϭ 7n
12. an ϭ 213nϪ1 2
For the given sum formula Sn, find S4, S5, Sk, and Sk؉1.
13. Sn ϭ n15n Ϫ 12
15. Sn ϭ
n1n ϩ 12
2
17. Sn ϭ 2n Ϫ 1
14. Sn ϭ n13n Ϫ 12
16. Sn ϭ
7n1n ϩ 12
2
18. Sn ϭ 3n Ϫ 1
Verify that S4 ؉ a5 ؍S5 for each exercise. Note that
each Sn is identical to those in Exercises 13 through 18.
19. an ϭ 10n Ϫ 6; Sn ϭ n15n Ϫ 12
20. an ϭ 6n Ϫ 4; Sn ϭ n13n Ϫ 12
21. an ϭ n; Sn ϭ
n1n ϩ 12
22. an ϭ 7n; Sn ϭ
23. an ϭ 2
nϪ1
2
7n1n ϩ 12
2
; Sn ϭ 2 Ϫ 1
n
24. an ϭ 213nϪ1 2; Sn ϭ 3n Ϫ 1
WORKING WITH FORMULAS
25. Sum of the first n cubes (alternative form):
(1 ؉ 2 ؉ 3 ؉ 4 ؉ p ؉ n)2
Earlier we noted the formula for the sum of the
n2 1n ϩ 12 2
. An alternative is
first n cubes was
4
given by the formula shown.
a. Verify the formula for n ϭ 1, 5, and 9.
ᮣ
4. The graph of a sequence is
, meaning it
is made up of distinct points.
6. Discuss the similarities and differences between
mathematical induction applied to sums and the
general principle of mathematical induction.
DEVELOPING YOUR SKILLS
For the given nth term an, find a4, a5, ak, and ak؉1.
ᮣ
2. Showing a statement is true for n ϭ 1 is called the
of an inductive proof.
b. Verify the formula using
1ϩ2ϩ3ϩpϩnϭ
n1n ϩ 12
2
.
26. Powers of the imaginary unit: in ؉ 4 ؍in, where
i ؍1 ؊1
Use a proof by induction to prove that powers of
the imaginary unit are cyclic. That is, that they
cycle through the numbers i, Ϫ1, Ϫi, and 1 for
consecutive powers.
APPLICATIONS
Use mathematical induction to prove the indicated sum
formula is true for all natural numbers n.
27. 2 ϩ 4 ϩ 6 ϩ 8 ϩ 10 ϩ p ϩ 2n;
an ϭ 2n, Sn ϭ n1n ϩ 12
28. 3 ϩ 7 ϩ 11 ϩ 15 ϩ 19 ϩ p ϩ 14n Ϫ 12;
an ϭ 4n Ϫ 1, Sn ϭ n12n ϩ 12
29. 5 ϩ 10 ϩ 15 ϩ 20 ϩ 25 ϩ p ϩ 5n;
5n1n ϩ 12
an ϭ 5n, Sn ϭ
2
30. 1 ϩ 4 ϩ 7 ϩ 10 ϩ 13 ϩ p ϩ 13n Ϫ 22;
n13n Ϫ 12
an ϭ 3n Ϫ 2, Sn ϭ
2
cob19545_ch09_797-805.qxd
11/9/10
9:38 PM
Page 802
College Algebra Graphs & Models—
802
9–42
CHAPTER 9 Additional Topics in Algebra
31. 5 ϩ 9 ϩ 13 ϩ 17 ϩ p ϩ 14n ϩ 12;
an ϭ 4n ϩ 1, Sn ϭ n12n ϩ 32
32. 4 ϩ 12 ϩ 20 ϩ 28 ϩ 36 ϩ p ϩ 18n Ϫ 42;
an ϭ 8n Ϫ 4, Sn ϭ 4n2
37.
1
1
1
1
ϩ
ϩ
ϩpϩ
;
1132
3152
5172
12n Ϫ 1212n ϩ 12
1
n
an ϭ
, Sn ϭ
2n ϩ 1
12n Ϫ 12 12n ϩ 12
1
1
1
1
ϩ
ϩ
ϩpϩ
;
1122
2132
3142
n1n ϩ 12
1
n
, Sn ϭ
an ϭ
n
ϩ
1
n1n ϩ 12
Use the principle of mathematical induction to prove
that each statement is true for all natural numbers n.
33. 3 ϩ 9 ϩ 27 ϩ 81 ϩ 243 ϩ p ϩ 3n;
313n Ϫ 12
an ϭ 3n, Sn ϭ
2
34. 5 ϩ 25 ϩ 125 ϩ 625 ϩ p ϩ 5n;
515n Ϫ 12
an ϭ 5n, Sn ϭ
4
35. 2 ϩ 4 ϩ 8 ϩ 16 ϩ 32 ϩ 64 ϩ p ϩ 2n;
an ϭ 2n, Sn ϭ 2nϩ1 Ϫ 2
38.
39. 3n Ն 2n ϩ 1
40. 2n Ն n ϩ 1
41. 3 # 4nϪ1 Յ 4n Ϫ 1
42. 4 # 5nϪ1 Յ 5n Ϫ 1
43. n2 Ϫ 7n is divisible by 2
36. 1 ϩ 8 ϩ 27 ϩ 64 ϩ 125 ϩ 216 ϩ p ϩ n3;
n2 1n ϩ 12 2
an ϭ n3, Sn ϭ
4
44. n3 Ϫ n ϩ 3 is divisible by 3
45. n3 ϩ 3n2 ϩ 2n is divisible by 3
46. 5n Ϫ 1 is divisible by 4
47. 6n Ϫ 1 is divisible by 5
ᮣ
EXTENDING THE CONCEPT
48. You may have noticed that the sum formula for the first n integers was quadratic, and the formula for the first n
integer squares was cubic. Is the formula for the first n integer cubes, if it exists, a quartic (degree four) function?
Use your calculator to run a quartic regression on the first five perfect cubes (enter 1 through 5 in L1 and the
cumulative sums in L2). What did you find? How is this exercise related to Exercise 36?
xn Ϫ 1
ϭ 11 ϩ x ϩ x2 ϩ x3 ϩ p ϩ xnϪ1 2.
xϪ1
50. Use mathematical induction to prove that for 14 ϩ 24 ϩ 34 ϩ p ϩ n4, where an ϭ n4,
49. Use mathematical induction to prove that
Sn ϭ
ᮣ
n1n ϩ 12 12n ϩ 12 13n2 ϩ 3n Ϫ 12
30
.
MAINTAINING YOUR SKILLS
51. (7.2) Given the matrices A ϭ c
Ϫ1
3
53. (1.1) State the equation of the circle whose graph
is shown here.
2
d and
1
2 Ϫ1
d , find A ϩ B, A Ϫ B, 2A Ϫ 3B,
4
3
AB, BA, and BϪ1.
Bϭ c
52. (2.5) State the domain and
range of the piecewise
function shown here.
y
10
(1, 7)
8
6
(4, 3)
4
2
Ϫ10Ϫ8Ϫ6Ϫ4Ϫ2
Ϫ2
Ϫ4
Ϫ6
Ϫ8
Ϫ10
y
5
4
3
(Ϫ1, 1) 2
1
Ϫ5Ϫ4Ϫ3Ϫ2Ϫ1
Ϫ1
Ϫ2
Ϫ3
Ϫ4
Ϫ5
1 2 3
(3, Ϫ2)
5 x
2 4 6 8 10 x
54. (5.5) Solve: 3e2xϪ1 ϩ 5 ϭ 17. Answer in exact
form.
cob19545_ch09_797-805.qxd
12/17/10
12:55 AM
Page 803
College Algebra Graphs & Models—
9–43
803
Reinforcing Basic Concepts
MID-CHAPTER CHECK
1. an ϭ 7n Ϫ 4
Find the number of terms in each series and then find
the sum. Verify results on a graphing calculator.
12. 2 ϩ 5 ϩ 8 ϩ 11 ϩ p ϩ 74
2. an ϭ n2 ϩ 3
13.
In Exercises 1 through 3, the nth term is given. Write
the first three terms of each sequence and find a9.
3. an ϭ 1Ϫ12 n 12n Ϫ 12
͚3
ϩ 32 ϩ 52 ϩ 72 ϩ p ϩ 31
2
14. For an arithmetic series, a3 ϭ Ϫ8 and a7 ϭ 4.
Find S10.
4
4. Evaluate the sum
1
2
nϩ1
15. For a geometric series, a3 ϭ Ϫ81 and a6 ϭ 3.
Find S10.
nϭ1
5. Rewrite using sigma notation.
1 ϩ 4 ϩ 7 ϩ 10 ϩ 13 ϩ 16
16. Identify a1 and the common ratio r. Then find an
expression for the general term an.
a. 2, 6, 18, 54, p
Match each formula to its correct description.
n1a1 ϩ an 2
6. Sn ϭ
7. an ϭ a1rnϪ1
2
a1
8. Sq ϭ
9. an ϭ a1 ϩ 1n Ϫ 12d
1Ϫr
a1 11 Ϫ rn 2
10. Sn ϭ
1Ϫr
a. sum of an infinite geometric series
1
b. 12, 14, 18, 16
,p
17. Find the number of terms in the series then compute
the sum. 541 ϩ 181 ϩ 16 ϩ p ϩ 812
18. Find the infinite sum (if it exists).
Ϫ49 ϩ 1Ϫ72 ϩ 1Ϫ12 ϩ 1Ϫ17 2 ϩ p
19. Barrels of toxic waste are stacked at a storage facility
in pyramid form, with 60 barrels in the first row, 59 in
the second row, and so on, until there are 10 barrels in
the top row. How many barrels are in the storage
facility? Verify results using a graphing calculator.
b. nth term formula for an arithmetic series
c. sum of a finite geometric series
d. summation formula for an arithmetic series
20. As part of a conditioning regimen, a drill sergeant
orders her platoon to do 25 continuous standing
broad jumps. The best of these recruits was able to
jump 96% of the distance from the previous jump,
with a first jump distance of 8 ft. Use a sequence/
series to determine the distance the recruit jumped
on the 15th try, and the total distance traveled by the
recruit after all 25 jumps. Verify results using a
graphing calculator.
e. nth term formula for a geometric series
11. Identify a1 and the common difference d. Then find
an expression for the general term an.
a. 2, 5, 8, 11, p
b. 32, 94, 3, 15
4,p
REINFORCING BASIC CONCEPTS
Applications of Summation
The properties of summation play a large role in the development of key ideas in a first semester calculus course,
and the following summation formulas are an integral part of these ideas. The first three formulas were verified in
Section 9.4, while proof of the fourth was part of Exercise 48 on page 802.
n
(1)
͚
n
c ϭ cn
iϭ1
(2)
͚
iϭ
iϭ1
n1n ϩ 12
2
n
(3)
͚
i2 ϭ
n1n ϩ 1212n ϩ 12
iϭ1
6
n
(4)
͚
i3 ϭ
iϭ1
n2 1n ϩ 12 2
4
To see the various ways they can be applied consider the following.
Illustration 1 ᮣ Over several years, the owner of Morgan’s LawnCare has noticed that the company’s monthly
profits (in thousands) can be approximated by the sequence an ϭ 0.0625n3 Ϫ 1.25n2 ϩ 6n, with the points plotted in
Figure 9.52 (the continuous graph is shown for effect only). Find the company’s approximate annual profit.
cob19545_ch09_797-805.qxd
11/9/10
9:39 PM
Page 804
College Algebra Graphs & Models—
804
9–44
CHAPTER 9 Additional Topics in Algebra
Solution ᮣ The most obvious approach would be to simply compute terms a1
through a12 (January through December) and find their sum: sum(seq(Y1, X,
1, 12), which gives a result of 35.75 or $35,750.
As an alternative, we could add the amount of profit earned by the company
in the first 8 months, then add the amount the company lost (or broke even) during
the last 4 months. In other words, we could apply summation property IV:
12
͚a
n
8
ϭ
Figure 9.52
12
0
12
͚a
12
n
ϩ
͚a
n
[(see Figure 9.53), which gives the same result:
42 ϩ 1Ϫ6.252 ϭ 35.75 or $35,750].
As a third option, we could use summation properties along with the
appropriate summation formulas, and compute the result manually. Note the
function is now written in terms of “i.” Distribute summation and factor out
constants (properties II and III):
iϭ1
iϭ1
12
͚
iϭ1
iϭ9
10.0625i3 Ϫ 1.25i2 ϩ 6i2 ϭ 0.0625
12
͚
12
i3 Ϫ 1.25
iϭ1
͚
Ϫ5
Figure 9.53
12
i2 ϩ 6
iϭ1
͚i
iϭ1
Replace each summation with the appropriate summation formula,
then substitute 12 for n:
ϭ 0.0625 c
ϭ 0.0625 c
n2 1n ϩ 12 2
4
2
1122 1132 2
d Ϫ 1.25 c
d Ϫ 1.25 c
n1n ϩ 12 12n ϩ 12
6
1122 1132 1252
4
6
ϭ 0.0625160842 Ϫ 1.2516502 ϩ 61782
ϭ 35.75
d ϩ 6c
d ϩ 6c
n1n ϩ 12
1122 1132
2
2
d
d
As we expected, the result shows profit was $35,750. While some approaches seem “easier” than others, all have
great value, are applied in different ways at different times, and are necessary to adequately develop key concepts in
future classes.
Exercise 1: Repeat Illustration 1 if the profit sequence is an ϭ 0.125x3 Ϫ 2.5x2 ϩ 12x.
9.5
Counting Techniques
LEARNING OBJECTIVES
In Section 9.5 you will see
how we can:
A. Count possibilities using
B.
C.
D.
E.
lists and tree diagrams
Count possibilities using
the fundamental principle
of counting
Quick-count
distinguishable
permutations
Quick-count
nondistinguishable
permutations
Quick-count using
combinations
How long would it take to estimate the number of fans sitting shoulder-to-shoulder at
a sold-out basketball game? Well, it depends. You could actually begin counting 1, 2,
3, 4, 5, p , which would take a very long time, or you could try to simplify the process
by counting the number of fans in the first row and multiplying by the number of rows.
Techniques for “quick-counting” the objects in a set or various subsets of a large set
play an important role in a study of probability.
A. Counting by Listing and Tree Diagrams
Consider the simple spinner shown in Figure 9.54, which is divided into three equal parts. What are the different possible outcomes for two spins, spin 1 followed by spin 2? We might begin by
organizing the possibilities using a tree diagram. As the name implies, each choice or possibility appears as the branch of a tree,
with the total possibilities being equal to the number of (unique)
Figure 9.54
B
A
C
cob19545_ch09_797-805.qxd
11/9/10
9:39 PM
Page 805
College Algebra Graphs & Models—
9–45
805
Section 9.5 Counting Techniques
Figure 9.55
paths from the beginning point to the end of a
branch. Figure 9.55 shows how the spinner exercise would appear (possibilities for two spins).
Moving from top to bottom we can trace nine
possible paths: AA, AB, AC, BA, BB, BC, CA,
CB, and CC.
Begin
A
A
EXAMPLE 1
ᮣ
B
B
C
A
B
C
C
A
B
C
Listing Possibilities Using a Tree Diagram
A basketball player is fouled and awarded three free throws. Let H represent the
possibility of a hit (basket is made), and M the possibility of a miss. Determine the
possible outcomes for the three shots using a tree diagram.
Solution
ᮣ
Each shot has two possibilities, hit (H) or miss (M), so the tree will branch in two
directions at each level. As illustrated in the figure, there are a total of eight
possibilities: HHH, HHM, HMH, HMM, MHH, MHM, MMH, and MMM.
Begin
H
M
H
H
M
M
H
H
M
H
M
M
H
M
Now try Exercises 7 through 10
WORTHY OF NOTE
Sample spaces may vary
depending on how we define the
experiment, and for simplicity’s
sake we consider only those
experiments having outcomes that
are equally likely.
ᮣ
To assist our discussion, an experiment is any task that can be done repeatedly and has
a well-defined set of possible outcomes. Each repetition of the experiment is called a
trial. A sample outcome is any potential outcome of a trial, and a sample space is a
set of all possible outcomes.
In our first illustration, the experiment was spinning a spinner, there were three
sample outcomes (A, B, or C), the experiment had two trials (spin 1 and spin 2), and
there were nine elements in the sample space. Note that after the first trial, each of the
three sample outcomes will again have three possibilities (A, B, and C). For two trials
we have 32 ϭ 9 possibilities, while three trials would yield a sample space with
33 ϭ 27 possibilities. In general, for N equally likely outcomes we have
A “Quick-Counting” Formula for a Sample Space
If an experiment has N sample outcomes that are equally likely and the experiment
is repeated t times, the number of elements in the sample space is
N t.
EXAMPLE 2
ᮣ
Counting the Outcomes in a Sample Space
Many combination locks have the digits 0 through 39 arranged
along a circular dial. Opening the lock requires stopping at a
sequence of three numbers within this range, going
counterclockwise to the first number, clockwise to the second,
and counterclockwise to the third. How many three-number
combinations are possible?
5
35
10
30
25
15
20