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  1. Preface
  2. 1 Prerequisites
    1. Introduction to Prerequisites
    2. 1.1 Real Numbers: Algebra Essentials
    3. 1.2 Exponents and Scientific Notation
    4. 1.3 Radicals and Rational Exponents
    5. 1.4 Polynomials
    6. 1.5 Factoring Polynomials
    7. 1.6 Rational Expressions
    8. Key Terms
    9. Key Equations
    10. Key Concepts
    11. Review Exercises
    12. Practice Test
  3. 2 Equations and Inequalities
    1. Introduction to Equations and Inequalities
    2. 2.1 The Rectangular Coordinate Systems and Graphs
    3. 2.2 Linear Equations in One Variable
    4. 2.3 Models and Applications
    5. 2.4 Complex Numbers
    6. 2.5 Quadratic Equations
    7. 2.6 Other Types of Equations
    8. 2.7 Linear Inequalities and Absolute Value Inequalities
    9. Key Terms
    10. Key Equations
    11. Key Concepts
    12. Review Exercises
    13. Practice Test
  4. 3 Functions
    1. Introduction to Functions
    2. 3.1 Functions and Function Notation
    3. 3.2 Domain and Range
    4. 3.3 Rates of Change and Behavior of Graphs
    5. 3.4 Composition of Functions
    6. 3.5 Transformation of Functions
    7. 3.6 Absolute Value Functions
    8. 3.7 Inverse Functions
    9. Key Terms
    10. Key Equations
    11. Key Concepts
    12. Review Exercises
    13. Practice Test
  5. 4 Linear Functions
    1. Introduction to Linear Functions
    2. 4.1 Linear Functions
    3. 4.2 Modeling with Linear Functions
    4. 4.3 Fitting Linear Models to Data
    5. Key Terms
    6. Key Concepts
    7. Review Exercises
    8. Practice Test
  6. 5 Polynomial and Rational Functions
    1. Introduction to Polynomial and Rational Functions
    2. 5.1 Quadratic Functions
    3. 5.2 Power Functions and Polynomial Functions
    4. 5.3 Graphs of Polynomial Functions
    5. 5.4 Dividing Polynomials
    6. 5.5 Zeros of Polynomial Functions
    7. 5.6 Rational Functions
    8. 5.7 Inverses and Radical Functions
    9. 5.8 Modeling Using Variation
    10. Key Terms
    11. Key Equations
    12. Key Concepts
    13. Review Exercises
    14. Practice Test
  7. 6 Exponential and Logarithmic Functions
    1. Introduction to Exponential and Logarithmic Functions
    2. 6.1 Exponential Functions
    3. 6.2 Graphs of Exponential Functions
    4. 6.3 Logarithmic Functions
    5. 6.4 Graphs of Logarithmic Functions
    6. 6.5 Logarithmic Properties
    7. 6.6 Exponential and Logarithmic Equations
    8. 6.7 Exponential and Logarithmic Models
    9. 6.8 Fitting Exponential Models to Data
    10. Key Terms
    11. Key Equations
    12. Key Concepts
    13. Review Exercises
    14. Practice Test
  8. 7 The Unit Circle: Sine and Cosine Functions
    1. Introduction to The Unit Circle: Sine and Cosine Functions
    2. 7.1 Angles
    3. 7.2 Right Triangle Trigonometry
    4. 7.3 Unit Circle
    5. 7.4 The Other Trigonometric Functions
    6. Key Terms
    7. Key Equations
    8. Key Concepts
    9. Review Exercises
    10. Practice Test
  9. 8 Periodic Functions
    1. Introduction to Periodic Functions
    2. 8.1 Graphs of the Sine and Cosine Functions
    3. 8.2 Graphs of the Other Trigonometric Functions
    4. 8.3 Inverse Trigonometric Functions
    5. Key Terms
    6. Key Equations
    7. Key Concepts
    8. Review Exercises
    9. Practice Test
  10. 9 Trigonometric Identities and Equations
    1. Introduction to Trigonometric Identities and Equations
    2. 9.1 Solving Trigonometric Equations with Identities
    3. 9.2 Sum and Difference Identities
    4. 9.3 Double-Angle, Half-Angle, and Reduction Formulas
    5. 9.4 Sum-to-Product and Product-to-Sum Formulas
    6. 9.5 Solving Trigonometric Equations
    7. Key Terms
    8. Key Equations
    9. Key Concepts
    10. Review Exercises
    11. Practice Test
  11. 10 Further Applications of Trigonometry
    1. Introduction to Further Applications of Trigonometry
    2. 10.1 Non-right Triangles: Law of Sines
    3. 10.2 Non-right Triangles: Law of Cosines
    4. 10.3 Polar Coordinates
    5. 10.4 Polar Coordinates: Graphs
    6. 10.5 Polar Form of Complex Numbers
    7. 10.6 Parametric Equations
    8. 10.7 Parametric Equations: Graphs
    9. 10.8 Vectors
    10. Key Terms
    11. Key Equations
    12. Key Concepts
    13. Review Exercises
    14. Practice Test
  12. 11 Systems of Equations and Inequalities
    1. Introduction to Systems of Equations and Inequalities
    2. 11.1 Systems of Linear Equations: Two Variables
    3. 11.2 Systems of Linear Equations: Three Variables
    4. 11.3 Systems of Nonlinear Equations and Inequalities: Two Variables
    5. 11.4 Partial Fractions
    6. 11.5 Matrices and Matrix Operations
    7. 11.6 Solving Systems with Gaussian Elimination
    8. 11.7 Solving Systems with Inverses
    9. 11.8 Solving Systems with Cramer's Rule
    10. Key Terms
    11. Key Equations
    12. Key Concepts
    13. Review Exercises
    14. Practice Test
  13. 12 Analytic Geometry
    1. Introduction to Analytic Geometry
    2. 12.1 The Ellipse
    3. 12.2 The Hyperbola
    4. 12.3 The Parabola
    5. 12.4 Rotation of Axes
    6. 12.5 Conic Sections in Polar Coordinates
    7. Key Terms
    8. Key Equations
    9. Key Concepts
    10. Review Exercises
    11. Practice Test
  14. 13 Sequences, Probability, and Counting Theory
    1. Introduction to Sequences, Probability and Counting Theory
    2. 13.1 Sequences and Their Notations
    3. 13.2 Arithmetic Sequences
    4. 13.3 Geometric Sequences
    5. 13.4 Series and Their Notations
    6. 13.5 Counting Principles
    7. 13.6 Binomial Theorem
    8. 13.7 Probability
    9. Key Terms
    10. Key Equations
    11. Key Concepts
    12. Review Exercises
    13. Practice Test
  15. A | Proofs, Identities, and Toolkit Functions
  16. Answer Key
    1. Chapter 1
    2. Chapter 2
    3. Chapter 3
    4. Chapter 4
    5. Chapter 5
    6. Chapter 6
    7. Chapter 7
    8. Chapter 8
    9. Chapter 9
    10. Chapter 10
    11. Chapter 11
    12. Chapter 12
    13. Chapter 13
  17. Index

Try It

13.1 Sequences and Their Notations

1.

The first five terms are { 1,6, 11, 16, 21 }. { 1,6, 11, 16, 21 }.

2.

The first five terms are { 2, 2,  3 2 , 1,  5 8 }. { 2, 2,  3 2 , 1,  5 8 }.

3.

The first six terms are { 2, 5, 54, 10, 250, 15 }. { 2, 5, 54, 10, 250, 15 }.

4.

a n = (1) n+1 9 n a n = (1) n+1 9 n

5.

a n = 3 n 4n a n = 3 n 4n

6.

a n = e n3 a n = e n3

7.

{ 2, 5, 11, 23, 47 } { 2, 5, 11, 23, 47 }

8.

{ 0, 1, 1, 1, 2, 3,  5 2 ,  17 6 }. { 0, 1, 1, 1, 2, 3,  5 2 ,  17 6 }.

9.

The first five terms are { 1,  3 2 , 4, 15, 72 }. { 1,  3 2 , 4, 15, 72 }.

13.2 Arithmetic Sequences

1.

The sequence is arithmetic. The common difference is 2. 2.

2.

The sequence is not arithmetic because 3163. 3163.

3.

{ 1 ,   6 ,   11 ,   16 ,   21 } { 1 ,   6 ,   11 ,   16 ,   21 }

4.

a 2 = 2 a 2 = 2

5.

a 1 = 25 a n = a n 1 + 12 ,  for  n 2 a 1 = 25 a n = a n 1 + 12 ,  for  n 2

6.

a n = 53 3 n a n = 53 3 n

7.

There are 11 terms in the sequence.

8.

The formula is T n =10+4n, T n =10+4n, and it will take her 42 minutes.

13.3 Geometric Sequences

1.

The sequence is not geometric because 10 5 15 10 10 5 15 10 .

2.

The sequence is geometric. The common ratio is 1 5 1 5 .

3.

{ 18,6,2, 2 3 , 2 9 } { 18,6,2, 2 3 , 2 9 }

4.

a 1 =2 a n = 2 3 a n1  for n2 a 1 =2 a n = 2 3 a n1  for n2

5.

a 6 =16,384 a 6 =16,384

6.

a n = (3) n1 a n = (3) n1

7.
  1. P n  = 2931.026 a n P n  = 2931.026 a n
  2. The number of hits will be about 333.

13.4 Series and Their Notations

1.

38

2.

26.4 26.4

3.

328 328

4.

−280 −280

5.

$2,025

6.

2,000.00 2,000.00

7.

9,840

8.

$275,513.31

9.

The sum is not defined.

10.

The sum of the infinite series is defined.

11.

The sum of the infinite series is defined.

12.

3

13.

The series is not geometric.

14.

3 11 3 11

15.

$32,775.87

13.5 Counting Principles

1.

7

2.

There are 60 possible breakfast specials.

3.

120

4.

60

5.

12

6.

P(7,7)=5,040 P(7,7)=5,040

7.

P(7,5)=2,520 P(7,5)=2,520

8.

C(10,3)=120 C(10,3)=120

9.

64 sundaes

10.

840

13.6 Binomial Theorem

1.
  1. 35
  2. 330
2.
  1. x 5 5 x 4 y+10 x 3 y 2 10 x 2 y 3 +5x y 4 y 5 x 5 5 x 4 y+10 x 3 y 2 10 x 2 y 3 +5x y 4 y 5
  2. 8 x 3 +60 x 2 y+150x y 2 +125 y 3 8 x 3 +60 x 2 y+150x y 2 +125 y 3
3.

10,206 x 4 y 5 10,206 x 4 y 5

13.7 Probability

1.
Outcome Probability
Heads 1212
Tails 1212
2.

2 3 2 3

3.

7 13 7 13

4.

2 13 2 13

5.

5 6 5 6

6.

a 1 91 ; b 5 91 ; c 86 91 a 1 91 ; b 5 91 ; c 86 91

13.1 Section Exercises

1.

A sequence is an ordered list of numbers that can be either finite or infinite in number. When a finite sequence is defined by a formula, its domain is a subset of the non-negative integers. When an infinite sequence is defined by a formula, its domain is all positive or all non-negative integers.

3.

Yes, both sets go on indefinitely, so they are both infinite sequences.

5.

A factorial is the product of a positive integer and all the positive integers below it. An exclamation point is used to indicate the operation. Answers may vary. An example of the benefit of using factorial notation is when indicating the product It is much easier to write than it is to write out 13121110987654321. 13121110987654321.

7.

First four terms: 8,  16 3 , 4,  16 5 8,  16 3 , 4,  16 5

9.

First four terms: 2,  1 2 ,  8 27 ,  1 4 2,  1 2 ,  8 27 ,  1 4 .

11.

First four terms: 1.25, 5, 20, 80 1.25, 5, 20, 80 .

13.

First four terms: 1 3 ,  4 5 ,  9 7 ,  16 9 1 3 ,  4 5 ,  9 7 ,  16 9 .

15.

First four terms: 4 5 , 4, 20, 100 4 5 , 4, 20, 100

17.

1 3 ,  4 5 ,  9 7 ,  16 9 ,  25 11 , 31, 44, 59 1 3 ,  4 5 ,  9 7 ,  16 9 ,  25 11 , 31, 44, 59

19.

0.6,3,15,20,375,80,9375,320 0.6,3,15,20,375,80,9375,320

21.

a n = n 2 +3 a n = n 2 +3

23.

a n = 2 n 2n  or  2 n1 n a n = 2 n 2n  or  2 n1 n

25.

a n = ( 1 2 ) n1 a n = ( 1 2 ) n1

27.

First five terms: 3, 9, 27, 81, 243 3, 9, 27, 81, 243

29.

First five terms: 1, 1, 9,  27 11 ,  891 5 1, 1, 9,  27 11 ,  891 5

31.

1 24 , 1,  1 4 ,  3 2 ,  9 4 ,  81 4 ,  2187 8 ,  531,441 16 1 24 , 1,  1 4 ,  3 2 ,  9 4 ,  81 4 ,  2187 8 ,  531,441 16

33.

2, 10, 12,  14 5 ,  4 5 , 2, 10, 12 2, 10, 12,  14 5 ,  4 5 , 2, 10, 12

35.

a 1 =8, a n = a n1 +n a 1 =8, a n = a n1 +n

37.

a 1 =35, a n = a n1 +3 a 1 =35, a n = a n1 +3

39.

720 720

41.

665,280 665,280

43.

First four terms: 1, 1 2 , 2 3 , 3 2 1, 1 2 , 2 3 , 3 2

45.

First four terms: 1,2, 6 5 , 24 11 1,2, 6 5 , 24 11

47.
Graph of a scattered plot with points at (1, 0), (2, 5/2), (3, 8/3), (4, 17/4), and (5, 24/5). The x-axis is labeled n and the y-axis is labeled a_n.
49.
Graph of a scattered plot with points at (1, 2), (2, 1), (3, 0), (4, 1), and (5, 0). The x-axis is labeled n and the y-axis is labeled a_n.
51.
Graph of a scattered plot with labeled points: (1, 2), (2, 6), (3, 12), (4, 20), and (5, 30). The x-axis is labeled n and the y-axis is labeled a_n.
53.

a n = 2 n2 a n = 2 n2

55.

a 1 =6,  a n =2 a n1 5 a 1 =6,  a n =2 a n1 5

57.

First five terms: 29 37 , 152 111 , 716 333 , 3188 999 , 13724 2997 29 37 , 152 111 , 716 333 , 3188 999 , 13724 2997

59.

First five terms: 2,3,5,17,65537 2,3,5,17,65537

61.

a 10 =7,257,600 a 10 =7,257,600

63.

First six terms: 0.042,0.146,0.875,2.385,4.708 0.042,0.146,0.875,2.385,4.708

65.

First four terms: 5.975,32.765,185.743,1057.25,6023.521 5.975,32.765,185.743,1057.25,6023.521

67.

If a n =421 a n =421 is a term in the sequence, then solving the equation 421=68n 421=68n for n n will yield a non-negative integer. However, if 421=68n, 421=68n, then n=51.875 n=51.875 so a n =421 a n =421 is not a term in the sequence.

69.

a 1 =1, a 2 =0, a n = a n1 a n2 a 1 =1, a 2 =0, a n = a n1 a n2

71.

(n+2)! (n1)! = (n+2)·(n+1)·(n)·(n1)·...·3·2·1 (n1)·...·3·2·1 =n(n+1)(n+2)= n 3 +3 n 2 +2n (n+2)! (n1)! = (n+2)·(n+1)·(n)·(n1)·...·3·2·1 (n1)·...·3·2·1 =n(n+1)(n+2)= n 3 +3 n 2 +2n

13.2 Section Exercises

1.

A sequence where each successive term of the sequence increases (or decreases) by a constant value.

3.

We find whether the difference between all consecutive terms is the same. This is the same as saying that the sequence has a common difference.

5.

Both arithmetic sequences and linear functions have a constant rate of change. They are different because their domains are not the same; linear functions are defined for all real numbers, and arithmetic sequences are defined for natural numbers or a subset of the natural numbers.

7.

The common difference is 1 2 1 2

9.

The sequence is not arithmetic because 1646416. 1646416.

11.

0, 2 3 , 4 3 ,2, 8 3 0, 2 3 , 4 3 ,2, 8 3

13.

0 , 5 , 10 , 15 , 20 0 , 5 , 10 , 15 , 20

15.

a 4 =19 a 4 =19

17.

a 6 =41 a 6 =41

19.

a 1 =2 a 1 =2

21.

a 1 =5 a 1 =5

23.

a 1 =6 a 1 =6

25.

a 21 =13.5 a 21 =13.5

27.

19,20.4,21.8,23.2,24.6 19,20.4,21.8,23.2,24.6

29.

a 1 =17;  a n = a n1 +9 n2 a 1 =17;  a n = a n1 +9 n2

31.

a 1 =12;  a n = a n1 +5 n2 a 1 =12;  a n = a n1 +5 n2

33.

a 1 =8.9;  a n = a n1 +1.4 n2 a 1 =8.9;  a n = a n1 +1.4 n2

35.

a 1 = 1 5 ;  a n = a n1 + 1 4 n2 a 1 = 1 5 ;  a n = a n1 + 1 4 n2

37.

1 = 1 6 ;  a n = a n1 13 12 n2 1 = 1 6 ;  a n = a n1 13 12 n2

39.

a 1 =4;  a n = a n1 +7;  a 14 =95 a 1 =4;  a n = a n1 +7;  a 14 =95

41.

First five terms: 20,16,12,8,4. 20,16,12,8,4.

43.

a n =1+2n a n =1+2n

45.

a n =105+100n a n =105+100n

47.

a n =1.8n a n =1.8n

49.

a n =13.1+2.7n a n =13.1+2.7n

51.

a n = 1 3 n 1 3 a n = 1 3 n 1 3

53.

There are 10 terms in the sequence.

55.

There are 6 terms in the sequence.

57.

The graph does not represent an arithmetic sequence.

59.
Graph of a scattered plot with labeled points: (1, 9), (2, -1), (3, -11), (4, -21), and (5, -31). The x-axis is labeled n and the y-axis is labeled a_n.
61.

1,4,7,10,13,16,19 1,4,7,10,13,16,19

63.
Graph of a scattered plot with labeled points: (1, 1), (2, 4), (3, 7), (4, 10), and (5, 13). The x-axis is labeled n and the y-axis is labeled a_n.
65.
Graph of a scattered plot with labeled points: (1, 5.5), (2, 6), (3, 6.5), (4, 7), and (5, 7.5). The x-axis is labeled n and the y-axis is labeled a_n.
67.

Answers will vary. Examples: a n =20.6n a n =20.6n and a n =2+20.4n. a n =2+20.4n.

69.

a 11 =17a+38b a 11 =17a+38b

71.

The sequence begins to have negative values at the 13th term, a 13 = 1 3 a 13 = 1 3

73.

Answers will vary. Check to see that the sequence is arithmetic. Example: Recursive formula: a 1 =3, a n = a n1 3. a 1 =3, a n = a n1 3. First 4 terms: 3,0,3,6 a 31 =87 3,0,3,6 a 31 =87

13.3 Section Exercises

1.

A sequence in which the ratio between any two consecutive terms is constant.

3.

Divide each term in a sequence by the preceding term. If the resulting quotients are equal, then the sequence is geometric.

5.

Both geometric sequences and exponential functions have a constant ratio. However, their domains are not the same. Exponential functions are defined for all real numbers, and geometric sequences are defined only for positive integers. Another difference is that the base of a geometric sequence (the common ratio) can be negative, but the base of an exponential function must be positive.

7.

The common ratio is 2 2

9.

The sequence is geometric. The common ratio is 2.

11.

The sequence is geometric. The common ratio is 1 2 . 1 2 .

13.

The sequence is geometric. The common ratio is 5. 5.

15.

5,1, 1 5 , 1 25 , 1 125 5,1, 1 5 , 1 25 , 1 125

17.

800,400,200,100,50 800,400,200,100,50

19.

a 4 = 16 27 a 4 = 16 27

21.

a 7 = 2 729 a 7 = 2 729

23.

7,1.4,0.28,0.056,0.0112 7,1.4,0.28,0.056,0.0112

25.

a = 1 32, a n = 1 2 a n1 a = 1 32, a n = 1 2 a n1

27.

a 1 =10, a n =0.3 a n1 a 1 =10, a n =0.3 a n1

29.

a 1 = 3 5 , a n = 1 6 a n1 a 1 = 3 5 , a n = 1 6 a n1

31.

a 1 = 1 512 , a n =4 a n1 a 1 = 1 512 , a n =4 a n1

33.

12,6,3, 3 2 , 3 4 12,6,3, 3 2 , 3 4

35.

a n = 3 n1 a n = 3 n1

37.

a n =0.8 (5) n1 a n =0.8 (5) n1

39.

a n = ( 4 5 ) n1 a n = ( 4 5 ) n1

41.

a n =3 ( 1 3 ) n1 a n =3 ( 1 3 ) n1

43.

a 12 = 1 177,147 a 12 = 1 177,147

45.

There are 12 12 terms in the sequence.

47.

The graph does not represent a geometric sequence.

49.
Graph of a scattered plot with labeled points: (1, 3), (2, 6), (3, 12), (4, 24), and (5, 48). The x-axis is labeled n and the y-axis is labeled a_n.
51.

Answers will vary. Examples: a 1 =800, a n =0.5a n1 a 1 =800, a n =0.5a n1 and a 1 =12.5, a n =4a n1 a 1 =12.5, a n =4a n1

53.

a 5 =256b a 5 =256b

55.

The sequence exceeds 100 100 at the 14th term, a 14 107. a 14 107.

57.

a 4 = 32 3 a 4 = 32 3 is the first non-integer value

59.

Answers will vary. Example: Explicit formula with a decimal common ratio: a n =400 0.5 n1 ; a n =400 0.5 n1 ; First 4 terms: 400,200,100,50; a 8 =3.125 400,200,100,50; a 8 =3.125

13.4 Section Exercises

1.

An nth nth partial sum is the sum of the first n n terms of a sequence.

3.

A geometric series is the sum of the terms in a geometric sequence.

5.

An annuity is a series of regular equal payments that earn a constant compounded interest.

7.

n=0 4 5n n=0 4 5n

9.

k=1 5 4 k=1 5 4

11.

k=1 20 8k+2 k=1 20 8k+2

13.

S 5 = 5( 3 2 + 7 2 ) 2 S 5 = 5( 3 2 + 7 2 ) 2

15.

S 13 = 13( 3.2+5.6 ) 2 S 13 = 13( 3.2+5.6 ) 2

17.

k=1 7 8 0.5 k1 k=1 7 8 0.5 k1

19.

S 5 = 9( 1 ( 1 3 ) 5 ) 1 1 3 = 121 9 13.44 S 5 = 9( 1 ( 1 3 ) 5 ) 1 1 3 = 121 9 13.44

21.

S 11 = 64( 1 0.2 11 ) 10.2 = 781,249,984 9,765,625 80 S 11 = 64( 1 0.2 11 ) 10.2 = 781,249,984 9,765,625 80

23.

The series is defined. S= 2 10.8 S= 2 10.8

25.

The series is defined. S= 1 1( 1 2 ) S= 1 1( 1 2 )

27.
Graph of Javier's deposits where the x-axis is the months of the year and the y-axis is the sum of deposits.
29.

Sample answer: The graph of S n S n seems to be approaching 1. This makes sense because k=1 ( 1 2 ) k k=1 ( 1 2 ) k is a defined infinite geometric series with S= 1 2 1( 1 2 ) =1. S= 1 2 1( 1 2 ) =1.

31.

49

33.

254

35.

S 7 = 147 2 S 7 = 147 2

37.

S 11 = 55 2 S 11 = 55 2

39.

S 7 =5208.4 S 7 =5208.4

41.

S 10 = 1023 256 S 10 = 1023 256

43.

S= 4 3 S= 4 3

45.

S=9.2 S=9.2

47.

$3,705.42

49.

$695,823.97

51.

a k =30k a k =30k

53.

9 terms

55.

r= 4 5 r= 4 5

57.

$400 per month

59.

420 feet

61.

12 feet

13.5 Section Exercises

1.

There are m+n m+n ways for either event A A or event B B to occur.

3.

The addition principle is applied when determining the total possible of outcomes of either event occurring. The multiplication principle is applied when determining the total possible outcomes of both events occurring. The word “or” usually implies an addition problem. The word “and” usually implies a multiplication problem.

5.

A combination; C(n,r)= n! (nr)!r! C(n,r)= n! (nr)!r!

7.

4+2=6 4+2=6

9.

5+4+7=16 5+4+7=16

11.

2×6=12 2×6=12

13.

10 3 =1000 10 3 =1000

15.

P(5,2)=20 P(5,2)=20

17.

P(3,3)=6 P(3,3)=6

19.

P(11,5)=55,440 P(11,5)=55,440

21.

C(12,4)=495 C(12,4)=495

23.

C(7,6)=7 C(7,6)=7

25.

2 10 =1024 2 10 =1024

27.

2 12 =4096 2 12 =4096

29.

2 9 =512 2 9 =512

31.

8! 3! =6720 8! 3! =6720

33.

12! 3!2!3!4! 12! 3!2!3!4!

35.

9

37.

Yes, for the trivial cases r=0 r=0 and r=1. r=1. If r=0, r=0, then C(n,r)=P(n,r)=1.  C(n,r)=P(n,r)=1.  If r=1, r=1, then r=1, r=1, C(n,r)=P(n,r)=n. C(n,r)=P(n,r)=n.

39.

6! 2! ×4!=8640 6! 2! ×4!=8640

41.

63+83=8 63+83=8

43.

4×2×5=40 4×2×5=40

45.

4×12×3=144 4×12×3=144

47.

P(15,9)=1,816,214,400 P(15,9)=1,816,214,400

49.

C(10,3)×C(6,5)×C(5,2)=7,200 C(10,3)×C(6,5)×C(5,2)=7,200

51.

2 11 =2048 2 11 =2048

53.

20! 6!6!8! =116,396,280 20! 6!6!8! =116,396,280

13.6 Section Exercises

1.

A binomial coefficient is an alternative way of denoting the combination C(n,r). C(n,r). It is defined as ( n r )=C(n,r)= n! r!(nr)! . ( n r )=C(n,r)= n! r!(nr)! .

3.

The Binomial Theorem is defined as (x+y) n = k=0 n ( n k ) x nk y k (x+y) n = k=0 n ( n k ) x nk y k and can be used to expand any binomial.

5.

15

7.

35

9.

10

11.

12,376

13.

64 a 3 48 a 2 b+12a b 2 b 3 64 a 3 48 a 2 b+12a b 2 b 3

15.

27 a 3 +54 a 2 b+36a b 2 +8 b 3 27 a 3 +54 a 2 b+36a b 2 +8 b 3

17.

1024 x 5 +2560 x 4 y+2560 x 3 y 2 +1280 x 2 y 3 +320x y 4 +32 y 5 1024 x 5 +2560 x 4 y+2560 x 3 y 2 +1280 x 2 y 3 +320x y 4 +32 y 5

19.

1024 x 5 3840 x 4 y+5760 x 3 y 2 4320 x 2 y 3 +1620x y 4 243 y 5 1024 x 5 3840 x 4 y+5760 x 3 y 2 4320 x 2 y 3 +1620x y 4 243 y 5

21.

1 x 4 + 8 x 3 y + 24 x 2 y 2 + 32 x y 3 + 16 y 4 1 x 4 + 8 x 3 y + 24 x 2 y 2 + 32 x y 3 + 16 y 4

23.

a 17 +17 a 16 b+136 a 15 b 2 a 17 +17 a 16 b+136 a 15 b 2

25.

a 15 30 a 14 b+420 a 13 b 2 a 15 30 a 14 b+420 a 13 b 2

27.

3,486,784,401 a 20 +23,245,229,340 a 19 b+73,609,892,910 a 18 b 2 3,486,784,401 a 20 +23,245,229,340 a 19 b+73,609,892,910 a 18 b 2

29.

x 24 8 x 21 y +28 x 18 y x 24 8 x 21 y +28 x 18 y

31.

720 x 2 y 3 720 x 2 y 3

33.

220,812,466,875,000 y 7 220,812,466,875,000 y 7

35.

35 x 3 y 4 35 x 3 y 4

37.

1,082,565 a 3 b 16 1,082,565 a 3 b 16

39.

1152 y 2 x 7 1152 y 2 x 7

41.

f 2 (x)= x 4 +12 x 3 f 2 (x)= x 4 +12 x 3

Graph of the function f_2.
43.

f 4 (x)= x 4 +12 x 3 +54 x 2 +108x f 4 (x)= x 4 +12 x 3 +54 x 2 +108x

Graph of the function f_4.
45.

590,625 x 5 y 2 590,625 x 5 y 2

47.

k1 k1

49.

The expression ( x 3 +2 y 2 z) 5 ( x 3 +2 y 2 z) 5 cannot be expanded using the Binomial Theorem because it cannot be rewritten as a binomial.

13.7 Section Exercises

1.

probability; The probability of an event is restricted to values between 0 0 and 1, 1, inclusive of 0 0 and 1. 1.

3.

An experiment is an activity with an observable result.

5.

The probability of the union of two events occurring is a number that describes the likelihood that at least one of the events from a probability model occurs. In both a union of sets A  and B A  and B and a union of events A and B, A and B, the union includes either A or B A or B or both. The difference is that a union of sets results in another set, while the union of events is a probability, so it is always a numerical value between 0 0 and 1. 1.

7.

1 2 . 1 2 .

9.

5 8 . 5 8 .

11.

1 2 . 1 2 .

13.

3 8 . 3 8 .

15.

1 4 . 1 4 .

17.

3 4 . 3 4 .

19.

3 8 . 3 8 .

21.

1 8 . 1 8 .

23.

15 16 . 15 16 .

25.

5 8 . 5 8 .

27.

1 13 . 1 13 .

29.

1 26 . 1 26 .

31.

12 13 . 12 13 .

33.
1 2 3 4 5 6
1 (1, 1)
2
(1, 2)
3
(1, 3)
4
(1, 4)
5
(1, 5)
6
(1, 6)
7
2 (2, 1)
3
(2, 2)
4
(2, 3)
5
(2, 4)
6
(2, 5)
7
(2, 6)
8
3 (3, 1)
4
(3, 2)
5
(3, 3)
6
(3, 4)
7
(3, 5)
8
(3, 6)
9
4 (4, 1)
5
(4, 2)
6
(4, 3)
7
(4, 4)
8
(4, 5)
9
(4, 6)
10
5 (5, 1)
6
(5, 2)
7
(5, 3)
8
(5, 4)
9
(5, 5)
10
(5, 6)
11
6 (6, 1)
7
(6, 2)
8
(6, 3)
9
(6, 4)
10
(6, 5)
11
(6, 6)
12
35.

5 12 . 5 12 .

37.

0. 0.

39.

4 9 . 4 9 .

41.

1 4 . 1 4 .

43.

5 8 5 8

45.

8 13 8 13

47.

C(12,5) C(48,5) = 1 2162 C(12,5) C(48,5) = 1 2162

49.

C(12,3)C(36,2) C(48,5) = 175 2162 C(12,3)C(36,2) C(48,5) = 175 2162

51.

C(20,3)C(60,17) C(80,20) 12.49% C(20,3)C(60,17) C(80,20) 12.49%

53.

C(20,5)C(60,15) C(80,20) 23.33% C(20,5)C(60,15) C(80,20) 23.33%

55.

20.50+23.3312.49=31.34% 20.50+23.3312.49=31.34%

57.

C(40000000,1)C(277000000,4) C(317000000,5) =36.78% C(40000000,1)C(277000000,4) C(317000000,5) =36.78%

59.

C(40000000,4)C(277000000,1) C(317000000,5) =0.11% C(40000000,4)C(277000000,1) C(317000000,5) =0.11%

Review Exercises

1.

2,4,7,11 2,4,7,11

3.

13,103,1003,10003 13,103,1003,10003

5.

The sequence is arithmetic. The common difference is d= 5 3 . d= 5 3 .

7.

18,10,2,6,14 18,10,2,6,14

9.

a 1 =20,  a n = a n1 +10 a 1 =20,  a n = a n1 +10

11.

a n = 1 3 n+ 13 24 a n = 1 3 n+ 13 24

13.

r=2 r=2

15.

4, 16, 64, 256, 1024 4, 16, 64, 256, 1024

17.

3, 12, 48, 192, 768 3, 12, 48, 192, 768

19.

a n = 1 5 ( 1 3 ) n1 a n = 1 5 ( 1 3 ) n1

21.

m=0 5 ( 1 2 m+5 ). m=0 5 ( 1 2 m+5 ).

23.

S 11 =110 S 11 =110

25.

S 9 23.95 S 9 23.95

27.

S= 135 4 S= 135 4

29.

$5,617.61

31.

6

33.

10 4 =10,000 10 4 =10,000

35.

P(18,4)=73,440 P(18,4)=73,440

37.

C( 15,6 )=5005 C( 15,6 )=5005

39.

2 50 =1.13× 10 15 2 50 =1.13× 10 15

41.

8! 3!2! =3360 8! 3!2! =3360

43.

490,314 490,314

45.

131,072 a 17 +1,114,112 a 16 b+4,456,448 a 15 b 2 131,072 a 17 +1,114,112 a 16 b+4,456,448 a 15 b 2

47.
1 2 3 4 5 6
1 1, 1 1, 2 1, 3 1, 4 1, 5 1, 6
2 2, 1 2, 2 2, 3 2, 4 2, 5 2, 6
3 3, 1 3, 2 3, 3 3, 4 3, 5 3, 6
4 4, 1 4, 2 4, 3 4, 4 4, 5 4, 6
5 5, 1 5, 2 5, 3 5, 4 5, 5 5, 6
6 6, 1 6, 2 6, 3 6, 4 6, 5 6, 6
49.

1 6 1 6

51.

5 9 5 9

53.

4 9 4 9

55.

1 C( 350,8 ) C( 500,8 ) 94.4% 1 C( 350,8 ) C( 500,8 ) 94.4%

57.

C( 150,3 )C( 350,5 ) C( 500,8 ) 25.6% C( 150,3 )C( 350,5 ) C( 500,8 ) 25.6%

Practice Test

1.

14,6,2,0 14,6,2,0

3.

The sequence is arithmetic. The common difference is d=0.9. d=0.9.

5.

a 1 =2,  a n = a n1 3 2 ;  a 22 = 67 2 a 1 =2,  a n = a n1 3 2 ;  a 22 = 67 2

7.

The sequence is geometric. The common ratio is r= 1 2 . r= 1 2 .

9.

a 1 =1,  a n = 1 2 a n 1 a 1 =1,  a n = 1 2 a n 1

11.

k=3 15 ( 3 k 2 5 6 k ) k=3 15 ( 3 k 2 5 6 k )

13.

S 7 =2604.2 S 7 =2604.2

15.

Total in account: $140,355.75; $140,355.75; Interest earned: $14,355.75 $14,355.75

17.

5×3×2×3×2=180 5×3×2×3×2=180

19.

C( 15,3 )=455 C( 15,3 )=455

21.

10! 2!3!2! =151,200 10! 2!3!2! =151,200

23.

429 x 14 16 429 x 14 16

25.

4 7 4 7

27.

5 7 5 7

29.

C( 14,3 )C( 26,4 ) C( 40,7 ) 29.2% C( 14,3 )C( 26,4 ) C( 40,7 ) 29.2%

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