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z z 2w w w 2 3 2 3. Using back-substitution, the solution is x 1, y 2, z 1, and w 3. Now try Exercise 51. 333202_0801.qxd 12/5/05 10:59 AM Page 578 578 Chapter 8 Matrices and Determinants The procedure for using Gaussian elimination with back-substitution is summarized below. Gaussian Elimination with Back-Substitutio... |
easily found without using back-substitution, as illustrated in Example 8. Section 8.1 Matrices and Systems of Equations 579 Gauss-Jordan Elimination With Gaussian elimination, elementary row operations are applied to a matrix to obtain a (row-equivalent) row-echelon form of the matrix. A second method of elimination,... |
of Solutions Solve the system. 2x 4y 3x 5y 2z 0 1 Solution 2 3 →1 3 0 →1 →1 0 → 2R1 3R1 R2 R2 R1 2R2.............................. The corresponding system of equations is x 5z 2. y 3z 1 x y Solving for and x 5z 2 z, in terms of you have y 3z 1. and To write a solution to the system that does not use any of the three ... |
...... 17 4 9 17 4 9 17 4 5 9 4 5 4 4 5 2 Solution This row-echelon form is different from that obtained in Example 4. The corresponding system of linear equations for this row-echelon matrix is x 3y y 4 3z 5. z 2 Using back-substitution on this system, you obtain the solution x 1, y 1, and z 2 which is the same soluti... |
EREQUISITE SKILLS REVIEW: Practice and review algebra skills needed for this section at www.Eduspace.com. In Exercises 1– 6, determine the order of the matrix. 1. 7 0 2 36 3 33 9 3. 5. 45 20 2. 4. 6 15 3 6 7 0 3 7 In Exercises 7–12, write the augmented matrix for the system of linear equations. 7. 9. 11. 3y 3y 4x x x 1... |
5. to R2. R3. R3. (e) Add R1. 28. Perform the sequence of row operations on the matrix. R2 What did the operations accomplish? 7 0 3 4 1 2 4 1 to R3 (a) Add (c) Add 3 times R4. R1 (b) Interchange R1 times R2 7 (e) Multiply (d) Add by and R4. R3. to R4. to R1 1 2. (f) Add the appropriate multiples of R2 to R1, R3, and ... |
4 34 5 7 5 10 3z 24 z 14 6 2 28 14 2x 6y 16 2x 3y 7 x y 2x 4y 5x 5y 2x 3y x 3y 2x 6y 2x y 2x 3x 2x 2y 7x 5y 2y 3y y x x x x z z 4x 8x 15 10 14 z 2y 2z y 4z y 3z 3 3y 7z 5 9y 15z 9 51. 53. 55. 57. 59. 61. 63. 65. 67. 68. 69. 52. 54. 56. 58. 60. 62. 64. 66. 1.5 z z z 2x 3x 3x 2x y y 2y y 2y 27 13 22 9 3 3z 2z z 2 5 4 14... |
78. (a) 79. (a) 80. (a) y z z y 5z z x 2y z x 3y 4z x 4y 5z x 3y z y 7z z y 6z z (b) (b) (b) (b) 6 16 3 11 4 2 27 54 8 19 18 4 y 6 8 3 11 4 2 y 3z z 3z z x y 2z x 4y x 6y z 15 x y 3z y 5z 42 z 8 15 14 4 y 2z z x 2y 0 x y 6 3x 2y 8 81. Use the system x 3y z 3 x 5y 5z 1 2x 6y 3z 8 to write two different matrices in row-... |
86. Finance A small software corporation borrowed $500,000 to expand its software line. Some of the money was borrowed at 9%, some at 10%, and some at 12%. Use a system of equations to determine how much was borrowed at each rate if the annual interest was $52,000 and the amount borrowed at 10% was times the amount bo... |
equation in part (a) to estimate in the year 2008. Is the estimate reasonable? Explain. 333202_0801.qxd 12/5/05 11:00 AM Page 586 586 Chapter 8 Matrices and Determinants Network Analysis In Exercises 91 and 92, answer the questions about the specified network. (In a network it is assumed that the total flow into each ... |
can be performed on an augmented matrix. 99. What is the relationship between the three elementary row operations performed on an augmented matrix and the operations that lead to equivalent systems of equations? 100. Writing In your own words, describe the difference between a matrix in row-echelon form and a matrix i... |
equal if their corresponding entries are equal. bij aij Example 1 Equality of Matrices Solve for a11 a21 a11, a12, a21, and a12 a22 2 3 in the following matrix equation. a22 1 0 © Royalty-Free/Corbis Solution Because two matrices are equal only if their corresponding entries are equal, you can conclude that a11 2, a12... |
of by A 333202_0802.qxd 12/5/05 10:57 AM Page 589 Section 8.2 Operations with Matrices 589 Exploration Consider matrices A, B, and C below. Perform the indicated operations and compare the results. Find b. Find B A. and then add C to c. Find the resulting matrix. Find B C, then add A to the resulting matrix. 2B, two r... |
C and be cdA cdA) 1A A cA B cA cB c dA cA dA 1. 2. 3. 4. 5. 6. Commutative Property of Matrix Addition Associative Property of Matrix Addition Associative Property of Scalar Multiplication Scalar Identity Property Distributive Property Distributive Property Note that the Associative Property of Matrix Addition allows ... |
matrix The algebra of real numbers and the algebra of matrices have many similarities. For example, compare the following solutions. Real Numbers (Solve for x.) Matrices (Solve for X.) The algebra of real numbers and the algebra of matrices also have important differences, which will be discussed later. Example 6 Solv... |
p... cip... cmp a11 a21 a31... ai1... am1 a12 a22 a32... ai2... am2 a13 a23 a33... ai3... am3............... a1n a2n a3n... ain... amn ai1b1j ai2b2j ai3b3j... ainbnj cij Example 7 Finding the Product of Two Matrices AB First, note that the product equal to the number of rows of Moreover, the product A find the entries ... |
. In Example 10, note that the two products are different. Even if BA are defined, matrix multiplication is not, in general, commutative. That is, for most matrices, This is one way in which the algebra of real numbers and the algebra of matrices differ. AB BA. and AB Properties of Matrix Multiplication Let and be matr... |
Jordan elimination on the augmented matrix AX B. A B to solve for the matrix X. Solution a. In matrix form, AX B, the system can be written as follows x1 x2 x3 4 4 2 b. The augmented matrix is formed by adjoining matrix B to matrix A......... Using Gauss-Jordan elimination, you can rewrite this equation as......... So,... |
ULARY CHECK: In Exercises 1–4, fill in the blanks. 1. Two matrices are ________ if all of their corresponding entries are equal. 2. When performing matrix operations, real numbers are often referred to as ________. 3. A matrix consisting entirely of zeros is called a ________ matrix and is denoted by ________. 4. The n... |
5/05 10:57 AM Page 598 598 Chapter 8 Matrices and Determinants 15. 16. 17. 18. 1 2 44 0 5 2 30 18 4 9 2 1 3 3 0 14 6 24 3 7 1 7 1 4 13 4 6 8 65 1 9 6 3 0 3 2 11 1 3 9 5 1 1 In Exercises 19–22, use the matrix capabilities of a graphing utility to evaluate the expression. Round your results to three decimal places, if ne... |
. Use the matrix capabilities of a graphing utility to verify your answer. 1 47 333202_0802.qxd 12/5/05 10:57 AM Page 599 36 48. 49. 0 4 50 In Exercises 51–58, (a) write the system of linear equations and (b) use Gauss-Jordan as a matrix equation, elimination on the augmented matrix to solve for the matrix [A B] AX B, ... |
Write a matrix i of each crop what each entry (b) Write a matrix A that represents the number of bushels that are shipped to each outlet State aij B that represents the profit per bushel of of the matrix of the matrix represents. j. bij each fruit. State what each entry represents. (c) Find the product matrix represen... |
profit from milk sales for the weekend. 68. Profit At a convenience store, the numbers of gallons of 87-octane, 89-octane, and 93-octane gasoline sold over the A. weekend are represented by Octane 87 89 A 580 560 860 840 420 1020 93 320 160 540 Friday Saturday Sunday The selling prices per gallon and the profits per g... |
.36, the HMO standard plan costs $1187.76 and the HMO Plus plan costs $1248.12. The plant expects the costs of the plans to change next year as follows. For individuals, the costs for the comprehensive, HMO standard, and HMO Plus plans will be $683.91, $463.10, and $499.27, respectively. For families, the costs for the... |
it is not necessarily true that Illustrate this using the following However, if and A B 82. Think About It then If ab 0, a 0 matrices such that A O B O. or matrices 83. Exploration Let and be unequal diagonal matrices of the same order. (A diagonal matrix is a square matrix in which each entry not on the main diagonal... |
1x a1b x a1b a1 The number definition of the multiplicative inverse of a matrix is similar. is called the multiplicative inverse of a because a1a 1. The A matrix and let be the Definition of the Inverse of a Square Matrix Let be an exists a matrix AA1 In A1 n n A1 A1A is called the inverse of The symbol such that n n ... |
12 1 0 1 0 0 1 0 1 Equating corresponding entries, you obtain two systems of linear equations. x11 x11 x12 x12 4x21 3x21 4x22 3x22 1 0 0 1 Linear system with two variables, x11 and x21. Linear system with two variables, x12 and x22. Solve the first system using elementary row operations to determine that 4 x11 and x21 ... |
to using elementary row operations on the A The result will be the matrix I A1. If this is 3. Check your work by multiplying to see that AA1 I A1A. 333202_0803.qxd 12/8/05 10:44 AM Page 605 Section 8.3 The Inverse of a Square Matrix 605 Example 3 Finding the Inverse of a Matrix Find the inverse of Solution Begin by ad... |
c b d is invertible if and only if A then inverse is given by ad bc 0. Moreover, if ad bc 0, the A1 1 ad bc d c ad bc The denominator will study determinants in the next section. b a. Formula for inverse of matrix A is called the determinant of the 2 2 matrix You A. Example 4 Finding the Inverse of a 2 2 Matrix If pos... |
as x 0.06x x y 0.075y z 0.095z 2z 10,000 730 0 x, where respectively. Use an inverse matrix to solve the system. represent the amounts invested in AAA, AA, and B bonds, and y, z Solution Begin by writing the system in the matrix form AX B. 1 0.06 1 1 0.075 0 1 0.095 2 x y z 10,000 730 0 Then, use Gauss-Jordan eliminat... |
it exists). 11 15 20. 23. 25. 26. 24. 22. 19. 21. 3 2 6 0 3 3 3 2 In Exercises 27–38, use the matrix capabilities of a graphing utility to find the inverse of the matrix (if it exists). 27. 29. 31 10 15 2 0 3 1 4 3 2 1 2 28. 30. 5 3 10 3 32 11 6 2 5 2 333202_0803.qxd 12/5/05 11:01 AM Page 609 Section 8.3 The Inverse o... |
possible) the system of linear equations. 53. 3x 4y 5x 3y 2 4 54. 18x 12y 13 30x 24y 23 55. 57. 59. 1.6 0.8y 4y 5 8 y 2 4 y 12 0.4x 2x 1 4 x 3 2 x 3 4x y z 2x 2y 3z 5x 2y 6z 3 5 10 1 56. 58. 60. 4 0.6y 1.4y y 20 2 y 51 0.2x x 5 6 x 3 x 4x 2y 3z 2x 2y 5z 8x 5y 2z 7 2.4 8.8 2 16 4 In Exercises 61–66, use the matrix capa... |
01 AM Page 610 610 Chapter 8 Matrices and Determinants 71. Circuit Analysis Consider the circuit shown in the figure. in amperes, are the solution of I2, I3, and I1, The currents the system of linear equations 4I3 4I3 I3 2I1 I2 I2 E1 E2 0 I1 E2 E1 and where are voltages. Use the inverse of the coefficient matrix of thi... |
. Exploration Consider matrices of the form A a11 0 0 0 0 a22 0 0 0 0 a33 ann (a) Write a A. matrix and a Find the inverse of each. 2 2 3 3 matrix in the form of (b) Use the result of part (a) to make a conjecture about the inverses of matrices in the form of A. Skills Review In Exercises 77 and 78, solve the inequalit... |
0. Coefficient Matrix A a1 a 2 b1 b2 Determinant detA a1b2 a 2b1 The determinant of the matrix can also be denoted by vertical bars on both sides of the matrix, as indicated in the following definition. A Definition of the Determinant of a 2 The determinant of the matrix 2 Matrix A a1 a2 b1 b2 is given by detA A a1 a ... |
5/05 11:03 AM Page 613 Section 8.4 The Determinant of a Square Matrix 613 Minors and Cofactors To define the determinant of a square matrix of order convenient to introduce the concepts of minors and cofactors. 3 3 or higher, it is Sign Pattern for Cofactors...... 3 3 matrix 4 4... n n matrix... matrix.................... |
A is A a11C11 a12C12... a1nC1n. Applying this definition to find a determinant is called expanding by cofactors. 2 2 matrix Try checking that for a A a1 a2 b1 b2 this definition of the determinant yields defined. A a1b2 a2b1, as previously Example 3 The Determinant of a Matrix of Order 3 3 Find the determinant of Solu... |
151 3 1 4 0 218 317 5. So, you obtain A 3C13 35 15. Now try Exercise 47. Try using a graphing utility to confirm the result of Example 4. 333202_0804.qxd 12/5/05 11:03 AM Page 616 616 Chapter 8 Matrices and Determinants 8.4 Exercises VOCABULARY CHECK: Fill in the blanks. A 1. Both Mij 2. The ________ detA and aij colum... |
indicated row or column. 31. 33. 3 2 5 3 (a) Row 1 4 2 (b) Column 2 3 4 3 0 12 6 5 0 1 1 6 1 32. 34. 3 6 4 4 3 7 (a) Row 2 2 1 8 (b) Column 3 5 0 10 10 30 0 5 10 1 (a) Row 2 (b) Column 2 (a) Row 3 (b) Column 1 35. 6 4 1 8 0 13 36. 10 a) Row 2 (b) Column 2 (a) Row 3 (b) Column 1 In Exercises 37–52, find the determinant... |
A to another column of B A. then A, A 1 1 2v 4u 3e3x e2x 1x x 2e2x ln x e3x 1 79. 81. 83. 1 3x 2 ex x 1 ex 3y 2 1 1 xex 1 ln x x ln x xex 80. 82. 84. Synthesis True or False? the statement is true or false. Justify your answer. In Exercises 85 and 86, determine whether 85. If a square matrix has an entire row of zeros... |
x x3 2x 96. gx 3x 97. hx 16 x2 99. gt lnt 1 98. 100. Ax 3 36 x2 f s 625e0.5s In Exercises 101 and 102, sketch the graph of the solution of the system of inequalities. 101. x x 2x y ≤ ≥ y < 8 3 5 102. x y > y ≤ 7x 4y ≤ 4 1 10 In Exercises 103–106, find the inverse of the matrix (if it exists). 103. 105 104. 5 3 106 333... |
c2 a1b2 a2c1 a2b1 c2 c1 a1 a2 b1 b2 b2 b1 a2 a1 a1 a2 c1 c2 b2 b1 Relative to the original system, the denominator for and is simply the deterD. minant of the coefficient matrix of the system. This determinant is denoted by respectively. They are The numerators for formed by using the column of constants as replacement... |
A Example 1 Using Cramer’s Rule for a 2 2 System Use Cramer’s Rule to solve the system of linear equations. 4x 2y 10 3x 5y 11 Because this determinant is not zero, you can apply Cramer’s Rule. Solution To begin, find the determinant of the coefficient matrix. D 4 3 x Dx D y Dy D 2 5 20 6 14 10 4 50 22 14 5 11 14 2 11 ... |
1, y1, x2, y2, and x3, y3 is x1 1 2 y1 y2 y3 1 1 1 Area ± x2 x3 ± where the symbol yield a positive area. indicates that the appropriate sign should be chosen to Example 3 Finding the Area of a Triangle (4, 3) Find the area of a triangle whose vertices are in Figure 8.1. 1, 0, 2, 2, and 4, 3, as shown Solution x1, y1 L... |
and x3, y3 are collinear (lie on the same line) Test for Collinear Points x1, y1 Three points if and only if, x2, y2, x1 x2 x3 y1 y2 y3 1 0. 1 1 Example 4 Testing for Collinear Points Determine whether the points Figure 8.3.) 2, 2, 1, 1, and 7, 5 are collinear. (See Solution Letting x1 x2 x3 x1, y1 y1 y2 y3 x3, y3 2, ... |
equation of the line is x 3y 10 0. Now try Exercise 39. Note that this method of finding the equation of a line works for all lines, including horizontal and vertical lines. For instance, the equation of the vertical line through 2, 2 2, 0 is x 2 2 1 1 and 1 0 y 0 2 4 2x 0 x 2. 333202_0805.qxd 12/5/05 11:05 AM Page 62... |
1 1 1 1 5 1 13 1 13 21 13 26 33 53 12 18 23 42 5 20 24 56 23 77 So, the sequence of coded row matrices is 13 26 21 33 53 12 18 23 42 5 20 56 24 23 77. Finally, removing the matrix notation produces the following cryptogram. 13 26 21 33 53 12 18 23 42 5 20 56 24 23 77 Now try Exercise 47. For those who do not know the ... |
a short paragraph describing how mathematics is used to code and decode messages. 333202_0805.qxd 12/5/05 11:05 AM Page 628 628 Chapter 8 Matrices and Determinants 8.5 Exercises VOCABULARY CHECK: Fill in the blanks. 1. The method of using determinants to solve a system of linear equations is called ________ ________. ... |
2y z 6 12. 14. 7 8 8 2x 2y 2z x 3y 4z x 2y z 2x 3y 5z 4 3x 5y 9z 7 5x 9y 17z 13 In Exercises 15–24, use a determinant and the given vertices of a triangle to find the area of the triangle. 15. y 16. y 5 4 3 2 1 (1, 5) (0, 0) (3, 14, 5) (0, 0) x −1 −2 4 1 (5, −2) −4 −2 2 4 x (−2, −3) (2, −3) (−2, 1) (3, −1) 2 −2 x 4 19... |
to find an equation of the line passing through the points. 39. 41. 43. 0, 0, 5, 3 4, 3, 2, 1 1 2, 1 2, 3, 5 40. 42. 44. 0, 0, 2, 2 10, 7, 2, 7 2 3, 4, 6, 12 In Exercises 45 and 46, find the uncoded 1 3 row matrices for the message. Then encode the message using the encoding matrix. Message 45. TROUBLE IN RIVER CITY E... |
5 2 1 3 19 28 9 19 80 25 5 4 112 140 83 19 25 13 72 118 71 20 21 38 35 23 36 42 48 32 76 61 95 333202_0805.qxd 12/5/05 11:05 AM Page 630 630 Chapter 8 Matrices and Determinants In Exercises 55 and 56, decode the cryptogram by using the inverse of the matrix A. Synthesis ] 55. 20 17 12 15 62 143 181 9 59 24 29 65 144 1... |
Skills Review In Exercises 63–66, use any method to solve the system of equations. 63. 64. 65. 66. 11 16 x 7y 22 5x y 26 3x 8y 2x 12y x 3y 5z 5x y z 4x 2y z 5x 3y 2z 2x 3y z 4x 10y 5z 14 1 11 7 5 37 In Exercises 67 and 68, sketch the region determined by the constraints. Then find the minimum and maximum values of the... |
a determinant to test for collinear points and to find an equation of a line passing through two points (p. 623). Use matrices to encode and decode messages (p. 625). Chapter Summary 631 Review Exercises 1–8 9, 10 11–24 25–30 31–34 35–48 49–62 63–66 67–70 71–78 79–82 83–94 95–98 99–102 103–106 107–110 111–114 115–120 ... |
y x y 0.3x 0.1y 0.13 0.2x 0.3y 0.25 0.2x 0.1y 0.07 0.4x 0.5y 0.01 2x 3y z 10 22 2 2x 3y 3z 3 2x y x 2y 6z 2x x 2x 5y 15z 3x y 3z z y 2y 3z 2z 3y z 2z 4 5 6z 2 2x 2y 2x y 2y 3y 4y 1 4 6 3x x 3z z z 4x 2x w 2w 3w w 3 0 2w 0 3 In Exercises 25–28, use matrices and Gauss-Jordan elimination to solve the system of equations. ... |
(c) and (d) A B, (b) A B, 4A, A 2 3 A 5 A 5 7 11 7 11 35. 36. 37. 38. A 6 5 A 3B 10 8 12 40 30 3 12 40 20 15 B 3 12 B 4 B 0 B 1 4 20 7, 4 8 In Exercises 39–42, perform the matrix operations. If it is not possible, explain why. 39. 7 1 40. 41. 11 7 21 5 6 10 3 14 5 19 1 87 16 2 2 4 0 20 10 A [4 1 3 0 5 2] 45. 47. X 3A ... |
. 7400 9800 4800 The price per unit is represented by the matrix B $10.25 $14.50 $17.75. Compute BA and interpret the result. 66. Long-Distance Plans The charges (in dollars per minute) of two long-distance telephone companies for in-state, stateto-state, and international calls are represented by C. Company A C 0.07 0... |
z 8x 4y z 6 1 7 12 25 10 13 11 0 14 8 44 79. 80. 81. 82. 83. 84. 85. 86. 87. 88. 89. 90. In Exercises 91–94, use the matrix capabilities of a graphing utility to solve (if possible) the system of linear equations. 91. 92. 93. x 2y 1 3x 4y 5 x 3y 6x 2y 3x 3y 4z y z 4x 3y 4z 23 18 2 1 1 Review Exercises 635 94. x 3y 2z 2... |
y Synthesis 8 6 4 2 −2 (5, 8) (5, 0) (1, 0) 4 6 8 113. (−2, 3) y 6 2 (0, 5) −4 −2 −2 −4 2 4 (1, −4) x x (0, 6) (4, 0) 6 2 −4 −2 (−4, 0) 2 4 114. y 3 2 1 3 2( (, 1 (4, 2) 1 2 3 1 4, − 2 ( ( x x In Exercises 115 and 116, use a determinant to determine whether the points are collinear. 1, 7, 3, 9, 3, 15 0, 5, 2, 6, 8, 1 ... |
ve the equation for. 2 3 8 0 5 333202_080R.qxd 12/5/05 11:08 AM Page 637 8 Chapter Test Chapter Test 637 Take this test as you would take a test in class. When you are finished, check your work against the answers given in the back of the book. In Exercises 1 and 2, write the matrix in reduced row-echelon form. 1. Writ... |
(a + c, b + d) (a, d) (a + c, d) (a, b) (0, 0) (a, 0) a c b d ad bc The following is a color-coded version of the proof along with a brief expla- nation of why this proof works. (0, d) (a, b + d) (a + c, b + d) (a, d) (a + c, d) (a, b) (0, 0) (a, 0) a c b d ad bc Area of Area of yellow Area of blue Area of orange Area... |
–17 12,589 15,886 25,916 5,226 14,906 Northeast Midwest South Mountain Pacific Northeast Midwest South Mountain Pacific 2000 18–64 33,175 39,486 62,235 11,210 28,036 2015 18–64 34,081 41,038 68,998 12,626 33,296 65 + 7,372 8,263 12,437 2,031 4,893 65 + 8,165 10,101 17,470 3,270 6,565 (a) The total population in 2000 wa... |
. Find an example of a singular 2 2 matrix satisfying A2 A. 639 333202_080R.qxd 12/5/05 11:08 AM Page 640 10. Verify the following equation. 9. Verify the following equation. a a2 a a3 1 1 1 b b2 1 b b3 1 c c2 a bb cc a c3 a bb cc aa b c a ax2 bx 11. Verify the following equation. 12. Use the equation given in Exercise... |
the encoding matrix Write and solve two systems y, of equations to find A. x,w, where A1 and z. (b) Decode the message. 18. Let. Use a graphing utility to find Make a conjecture about the determinant of the inverse of a matrix. Compare A1 A1. with n n 19. Let A zero. Find be an A. 20. Consider matrices of the form mat... |
5/05 11:28 AM Page 642 642 Chapter 9 Sequences, Series, and Probability 9.1 Sequences and Series What you should learn • Use sequence notation to write the terms of sequences. • Use factorial notation. • Use summation notation to write sums. • Find the sums of infinite series. • Use sequences and series to model and so... |
1 2 3 14 3 1 4. a2 a3 a4 1st term 2nd term 3rd term 4th term Now try Exercise 1. 333202_0901.qxd 12/5/05 11:28 AM Page 643 Section 9.1 Sequences and Series 643 Example 2 A Sequence Whose Terms Alternate in Sign Write the first five terms of the sequence given by an 1n 2n 1. Solution The first five terms of the sequenc... |
:28 AM Page 644 644 Chapter 9 Sequences, Series, and Probability Some sequences are defined recursively. To define a sequence recursively, you need to be given one or more of the first few terms. All other terms of the sequence are then defined using previous terms. A well-known example is the Fibonacci sequence shown ... |
. n whereas 2n! 1 2 3 4... 2n. Example 5 Writing the Terms of a Sequence Involving Factorials Write the first five terms of the sequence given by an. 2n n! n 0. Begin with Then graph the terms on a set of coordinate axes. Solution a0 a1 a 2 a3 a4 20 0! 21 1! 22 2! 23 3! 24 4 16 24 2 3 0th term 1st term 2nd term 3rd ter... |
lower limit of summation. n is the upper limit of Example 7 Summation Notation for Sums Find each sum. a. 5 i1 3i b. 6 k3 1 k2 c. 8 i0 1 i! Solution 5 a. i1 3i 31 32 33 34 35 31 2 3 4 5 315 45 b. 6 k3 1 k 2 1 32 1 42 1 52 1 62 10 17 26 37 90 c. 8 i0 1 i! 1 0! 1 1! 1 2! 1 6 1 3! 1 24 1 4! 1 120 1 1 6! 5! 1 720 1 7! 1 1... |
sequence and is denoted by a1 a2 a3... an n i1 ai. 2. The sum of all the terms of the infinite sequence is called an infinite series and is denoted by a1 a2 a3... ai... ai. i1 Example 8 Finding the Sum of a Series For the series i1 3 10i, find (a) the third partial sum and (b) the sum. Solution a. The third partial su... |
determine how cube have 0 blue faces, 1 blue face, 2 blue many unit cubes of the faces, and 3 blue faces. Do the same for a cube, and a pattern do you observe in the table? Write a formula you could use to determine the column values for an cube and add your results to the table below. What type of cube, a cube. Numbe... |
3. 5. an an an 7. an 9. an 11. an 13. an 15. an 17. an 19. 21. an an 3n 1 2n 2n n 2 n 6n 3n 2 1 1 1n n 2 1 3n 1 n32 1n n2 2 3 nn 1n 2 2. 4. 6. an an an 8. an 10. an 5n 3n2 n 4 2n2 1 12. an 1 1n 14. an 16. an 18. an 20. 22. an an 2n 3n 10 n23 1n n n 1 0.3 nn2 6 23. an a25 In Exercises 23–26, find the indicated term of ... |
. 48. 49. 50. 1 1 1 1 1, 1 1 2, 1 3 2, 1 1 4, 1 7 3, 1 1 8, 1 15 4, 1 1 16, 1 31 3, 7, 11, 15, 19,... 2, 4, 6, 8, 10,... 1 1 16,... 2, 1 3, 2 81,... 24, 1 1, 1 120,... 23 24 25 6 120 24 1 4, 1 8, 27, 8 9, 4 6, 1 2, 1 22 2 5,... 32,... 1, 2,,,,,... In Exercises 51–54, write the first five terms of the sequence defined r... |
i1 3i 1 5 5 k1 5 i0 2i 2 5 j3 1 j 2 3 4 i1 i 12 i 13 4 j0 2 j In Exercises 85–88, use a calculator to find the sum. 86. 10 j1 3 j 1 85. 87. 88. 6 j1 24 3j 4 k0 4 k0 1k k 1 1k k! In Exercises 89–98, use sigma notation to write the sum. 89. 90. 91. 92. 93. 94. 95. 96. 97. 98.... 1 39... 22 1 32 5 1 33 5 1 31 27 81 243 7... |
b) Find the balance in this account after 5 years by finding the 60th term of the sequence. (c) Find the balance in this account after 20 years by finding the 240th term of the sequence. Model It 109. Data Analysis: Number of Stores The table shows of Best Buy stores for the years 1998 the numbers to 2003. an (Source: ... |
the sequence. (b) What does the pattern in the bar graph in part (a) say about the future of the federal debt? 333202_0901.qxd 12/5/05 11:28 AM Page 652 652 Chapter 9 Sequences, Series, and Probability 112. Revenue The revenues an (in millions of dollars) for Amazon.com for the years 1996 through 2003 are shown in the... |
j2 j1 j3 Fibonacci Sequence Fibonacci sequence. (See Example 4.) In Exercises 115 and 116, use the 115. Write the first 12 terms of the Fibonacci sequence the first 10 terms of the sequence given by an and bn an1 an, n ≥ 1. 129. 130. 131. 116. Using the definition for bn in Exercise 115, show that bn 132. can be defin... |
. Example 1 Examples of Arithmetic Sequences a. The sequence whose n th term is 4n 3 is arithmetic. For this sequence, the common difference between consecutive terms is 4. 7, 11, 15, 19,..., 4n 3,... Begin with n 1. 11 7 4 b. The sequence whose n th term is 7 5n common difference between consecutive terms is 5. is ari... |
d Alternative form Example 2 Finding the nth Term of an Arithmetic Sequence Find a formula for the difference is 3 and whose first term is 2. n th term of the arithmetic sequence whose common Solution Because the sequence is arithmetic, you know that the formula for the th term d 3, an is of the form the formula must ... |
.... Now try Exercise 37. If you know the th term of an arithmetic sequence and you know the n th term by using the n 1 common difference of the sequence, you can find the recursion formula an1 an d. Recursion formula With this formula, you can find any term of an arithmetic sequence, provided that you know the precedi... |
only look at him in astounded silence. This is what Gauss did:... 100 3 2 1 100 101 99 101 98 101... 1... 101 Sn Sn 2Sn Sn 100 101 2 5050 520 100. Simplify. Now try Exercise 63. Example 6 Finding the Sum of a Finite Arithmetic Sequence Find the sum of the integers (a) from 1 to 100 and (b) from 1 to N. Solution a. The... |
50 1050 n th term of the sequence is is term sequence the of and the total amount of prize money is So, the you can determine that the formula for the an a16 50n 1050. 5016 1050 250, S16 1000 950 900... 250 n 2 a16 16th a1 S16 nth partial sum formula 16 2 1000 250 Substitute 16 for n, 1000 for a1, and 250 for a16. 812... |
40,000 20,000 ) FIGURE 9.4 333202_0902.qxd 12/5/05 11:30 AM Page 659 Section 9.2 Arithmetic Sequences and Partial Sums 659 9.2 Exercises VOCABULARY CHECK: Fill in the blanks. 1. A sequence is called an ________ sequence if the differences between two consecutive terms are the same. This difference is called the ______... |
2n n 17. an 18. an In Exercises 19–30, find a formula for sequence. an for the arithmetic 19. 20. 21. 22. 23. 24. 25. 26. 27. 1, d 3 15, d 4 100, d 8 0, d 2 3 x, d 2x y, d 5y 2, 1, 7 2,... a1 a1 a1 a1 a1 a1 4, 3 10, 5, 0, 5, 10,... a1 5, a4 15 31. 32. 33. 34. 35. 36. 37. 38. a1 a1 a1 a1 a1 a4 a8 a3 5, d 6 5, d 3 4 2.6... |
49. 51. an an 3 2 3 4 n 8 4 n 50. 52. an an 3n 5 25 3n 71. 73. 30 n11 n 10 n1 n 400 n1 2n 1 72. 74. 100 n51 n 50 n1 n 250 n1 1000 n In Exercises 75–80, use a graphing utility to find the partial sum. 20 1000 5n 2n 5 50 76. 75. n1 100 n1 60 i1 n 4 2 3i 250 8 77. 79. n0 78. 80. 100 n0 8 3n 16 200 j1 4.5 0.025j Job Offer... |
30 rows of seats if there are 20 seats in the first row, 24 seats in the second row, 28 seats in the third row, and so on. 84. Seating Capacity Determine the seating capacity of an auditorium with 36 rows of seats if there are 15 seats in the first row, 18 seats in the second row, 21 seats in the third row, and so on.... |
plus 1% interest on the unpaid balance. (a) Find the first year’s monthly payments you will make, and the unpaid balance after each month. (b) Find the total amount of interest paid over the term of the loan. Model It 95. Data Analysis: Personal Income The table shows in the United States (Source: U.S. Bureau of the p... |
13 $31,633 (a) Find an arithmetic sequence that models the data. corresponding represent the year, with n 3 n Let to 1993. (b) Use the regression feature of a graphing utility to find a linear model for the data. How does this model compare with the arithmetic sequence you found in part (a)? (c) Use a graphing utility ... |
first metic sequence with first term Sn. d Explain. a1 n Synthesis Skills Review True or False? the statement is true or false. Justify your answer. In Exercises 97 and 98, determine whether In Exercises 105–108, find the slope and y-intercept (if possible) of the equation of the line. Sketch the line. 97. Given an ar... |
Corbis Geometric Sequences In Section 9.2, you learned that a sequence whose consecutive terms have a common difference is an arithmetic sequence. In this section, you will study another important type of sequence called a geometric sequence. Consecutive terms of a geometric sequence have a common ratio. Definition of ... |
sequence. So, every where geometric sequence can be written in the following form. a1, a2, a3, a4, a5,....., an,..... a1, a1r, a1r2, a1r3, a1r 4,..., a1rn1,... n th term of a geometric sequence, you can find the If you know the ran. term by multiplying by That is, an1 r. n 1th Example 2 Finding the Terms of a Geometri... |
a geometric sequence is 125, and the 10th term is Find the 14th term. (Assume that the terms of the sequence are positive.) 12564. Solution The 10th term is related to the fourth term by the equation r is the common Remember that ratio of consecutive terms of a geometric sequence. So, in Example 5, a10 a1r 9 a1 r r r ... |
for a1, 0.3 for r, and 12 for n. Use a calculator. Now try Exercise 57. When using the formula for the sum of a finite geometric sequence, be careful to check that the sum is of the form n i1 a1 r i1. Exponent for r is i 1. If the sum is not of this form, you must adjust the formula. For instance, if the sum in Exampl... |
.6n 1... 4 1 0.6 10 a 1 1 r b. 3 0.3 0.03 0.003... 3 30.1 30.12 30.13... 3 1 0.1 a 1 1 r 10 3 3.33 Now try Exercise 79. 333202_0903.qxd 12/5/05 11:32 AM Page 668 668 Chapter 9 Sequences, Series, and Probability Application Example 8 Increasing Annuity Recall from Section 3.1 that the formula for compound interest is A ... |
able to make? Construct a sequence of the resulting string lengths after each cut, starting with the original length of the string. Find a formula for the nth term of this sequence. How many cuts could you theoretically make? Discuss why you were not able to make that many cuts. 333202_0903.qxd 12/5/05 11:32 AM Page 6... |
1 a1 20. a1 5, r 2x In Exercises 21–26, write the first five terms of the geometric sequence. Determine the common ratio and write the th term of the sequence as a function of n 21. 23. 25. a1 a1 a1 64, ak1 7, ak1 6, ak1 1 2ak 2a k 3 2ak 22. 24. 26. n. 81, ak1 5, ak1 48, ak1 a1 a1 a1 1 3ak 2ak 1 2 ak In Exercises 27–34... |
, Series, and Probability 47. In Exercises 47–52, use a graphing utility to graph the first 10 terms of the sequence. 120.75n1 120.4n1 21.3n1 101.5n1 201.25n1 101.2n1 51. 49. 50. 48. 52. an an an an an an In Exercises 53–72, find the sum of the finite geometric sequence. 9 2 n1 n1 10 53. 54. 5 2 n1 53 2 8 n1 n1 i1 21 4... |
Exercises 73–78, use summation notation to write the sum. 73. 74. 75. 76. 77. 78. 5 15 45... 3645 7 14 28... 896 2 1 1 8 2 15 3 3 5... 1 2048.. 3 625. 0.1 0.4 1.6... 102.4 32 24 18... 10.125 In Exercises 79–92, find the sum of the infinite geometric series. Year 1998 1999 2000 2001 2002 2003 2004 Population, an 1250.4... |
.) Find the depreciated value of the machine after 5 full years. 103. Annuities A deposit of $100 is made at the beginning of each month in an account that pays 6%, compounded monthly. The balance in the account at the end of 5 years is A 1001 0.06 12... 1001 0.06 12 60. 1 A Find A. 104. Annuities A deposit of $50 is m... |
account for an individual who retires at age 65 and wants an income of $2000 from the account each month for 20 years. Use the result of Exercise 111 and assume that the account earns 9% compounded monthly. Multiplier Effect In Exercises 113–116, use the following information. A tax rebate has been given to property o... |
) for each fall. s1 s2 s3 s4 16t 2 16, 16t 2 160.81, 16t 2 160.812, 16t 2 160.813,... 16t 2 160.81n1, sn 0 if t 1 0 if t 0.9 0 if t 0.9 2 0 if t 0.93... 0 if t 0.9n1 s1 s2 s3 s4 sn s2, the ball takes the same amount of Beginning with time to bounce up as it does to fall, and so the total time elapsed before it comes to... |
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