BBEPMC0R_0312279093.QXP 12/2/04 2:42 PM Page Basic Concepts of Algebra R.1 R.2 R.3 R.4 R.5 R.6 R.7 The Real-Number System Integer Exponents, Scientific Notation, and Order of Operations Addition, Subtraction, and Multiplication of Polynomials Factoring Rational Expressions Radical Notation and Rational Exponents The Basics of Equation Solving R SUMMARY AND REVIEW TEST A P P L I C A T I O N G ina wants to establish a college fund for her newborn daughter that will have accumulated $120,000 at the end of 18 yr If she can count on an interest rate of 6%, compounded monthly, how much should she deposit each month to accomplish this? This problem appears as Exercise 95 in Section R.2 Copyright © 2006 Pearson Education, Inc., publishing as Pearson Addison-Wesley BBEPMC0R_0312279093.QXP Chapter R 12/2/04 2:42 PM Page • Basic Concepts of Algebra Identify various kinds of real numbers Use interval notation to write a set of numbers Identify the properties of real numbers Find the absolute value of a real number R.1 2.1 The Real-Number Polynomial Functions System and Modeling Real Numbers In applications of algebraic concepts, we use real numbers to represent quantities such as distance, time, speed, area, profit, loss, and temperature Some frequently used sets of real numbers and the relationships among them are shown below Natural numbers (positive integers): 1, 2, 3, … Whole numbers: 0, 1, 2, 3, … Integers: …, −3, −2, −1, 0, 1, 2, 3, … Rational numbers Rational numbers that are not integers: Real numbers Negative integers: −1, −2, −3, … 19 −7 −, − −, −−, −−, 8.3, −5 − Irrational numbers: Zero: 0.56, … √2, p, −√3, √27, −4.030030003…, … Numbers that can be expressed in the form p͞q, where p and q are integers and q 0, are rational numbers Decimal notation for rational numbers either terminates (ends) or repeats Each of the following is a rational number a) b) Ϫ7 0.25 d) Ϫ Ϫ0.45 11 c) for any nonzero integer a a ؊7 ؊7 ؍ , or ؊1 0؍ Terminating decimal Repeating decimal Copyright © 2006 Pearson Education, Inc., publishing as Pearson Addison-Wesley BBEPMC0R_0312279093.QXP 12/2/04 2:42 PM Page Section R.1 • The Real-Number System The real numbers that are not rational are irrational numbers Decimal notation for irrational numbers neither terminates nor repeats Each of the following is an irrational number There is no repeating block of digits a) 3.1415926535 22 ͑ and 3.14 are rational approximations of the irrational number ͒ b) ͙2 1.414213562 c) Ϫ6.12122122212222 There is no repeating block of digits Although there is a pattern, there is no repeating block of digits The set of all rational numbers combined with the set of all irrational numbers gives us the set of real numbers The real numbers are modeled using a number line, as shown below Each point on the line represents a real number, and every real number is represented by a point on the line Ϫ2.9 ϪE Ϫ5 Ϫ4 Ϫ3 Ϫ2 Ϫ1 ͙3 * p The order of the real numbers can be determined from the number line If a number a is to the left of a number b, then a is less than b ͑a Ͻ b͒ Similarly, a is greater than b ͑a Ͼ b͒ if a is to the right of b on the number line For example, we see from the number line above that 17 Ϫ2.9 Ͻ Ϫ 35 , because Ϫ2.9 is to the left of Ϫ 35 Also, Ͼ ͙3, because 174 is to the right of ͙3 The statement a Յ b, read “a is less than or equal to b,” is true if either a Ͻ b is true or a b is true The symbol ʦ is used to indicate that a member, or element, belongs to a set Thus if we let ޑrepresent the set of rational numbers, we can see from the diagram on page that 0.56 ʦ ޑ We can also write ͙2 ޑto indicate that ͙2 is not an element of the set of rational numbers When all the elements of one set are elements of a second set, we say that the first set is a subset of the second set The symbol ʕ is used to denote this For instance, if we let ޒrepresent the set of real numbers, we can see from the diagram that ޑʕ ( ޒread “ ޑis a subset of )”ޒ Interval Notation Sets of real numbers can be expressed using interval notation For example, for real numbers a and b such that a Ͻ b, the open interval ͑a, b͒ is the set of real numbers between, but not including, a and b That is, ( a [ a ) (a, b) [a, ∞) b ͑a, b͒ ͕x ͉ a Ͻ x Ͻ b͖ The points a and b are endpoints of the interval The parentheses indicate that the endpoints are not included in the interval Some intervals extend without bound in one or both directions The interval ͓a, ϱ ͒, for example, begins at a and extends to the right without bound That is, ͓a, ϱ ͒ ͕x ͉ x Ն a͖ The bracket indicates that a is included in the interval Copyright © 2006 Pearson Education, Inc., publishing as Pearson Addison-Wesley BBEPMC0R_0312279093.QXP Chapter R 12/2/04 2:42 PM Page • Basic Concepts of Algebra The various types of intervals are listed below Intervals: Types, Notation, and Graphs TYPE INTERVAL NOTATION SET NOTATION Open ͑a, b͒ ͕x ͉ a Ͻ x Ͻ b͖ Closed Half-open Half-open Open ͓a, b͔ ͓a, b͒ ͑a, b͔ ͑a, ϱ ͒ GRAPH ͕x ͉ a Յ x Յ b͖ ͕x ͉ a Յ x Ͻ b͖ ͕x ͉ a Ͻ x Յ b͖ ͕x ͉ x Ͼ a͖ ( ) a b [ ] a b [ ) a b ( ] a b ( a Half-open ͓a, ϱ ͒ ͕x ͉ x Ն a͖ [ a Open ͑Ϫϱ, b͒ ͕x ͉ x Ͻ b͖ ) b Half-open ͑Ϫϱ, b͔ ͕x ͉ x Յ b͖ ] b The interval ͑Ϫϱ, ϱ ͒, graphed below, names the set of all real numbers, ޒ EXAMPLE Write interval notation for each set and graph the set a) ͕x ͉ Ϫ4 Ͻ x Ͻ 5͖ c) ͕x ͉ Ϫ5 Ͻ x Յ Ϫ2͖ b) ͕x ͉ x Ն 1.7͖ d) ͕ x ͉ x Ͻ ͙5 ͖ Solution a) ͕x ͉ Ϫ4 Ͻ x Ͻ 5͖ ͑Ϫ4, 5͒; Ϫ5 Ϫ4 Ϫ3 Ϫ2 Ϫ1 3 5 b) ͕x ͉ x Ն 1.7͖ ͓1.7, ϱ ͒; Ϫ5 Ϫ4 Ϫ3 Ϫ2 Ϫ1 c) ͕x ͉ Ϫ5 Ͻ x Յ Ϫ2͖ ͑Ϫ5, Ϫ2͔; Ϫ5 Ϫ4 Ϫ3 Ϫ2 Ϫ1 Copyright © 2006 Pearson Education, Inc., publishing as Pearson Addison-Wesley BBEPMC0R_0312279093.QXP 12/2/04 2:42 PM Page Section R.1 d) ͕ x ͉ x Ͻ ͙5 ͖ ͑ Ϫϱ, ͙5 ͒; • The Real-Number System Ϫ5 Ϫ4 Ϫ3 Ϫ2 Ϫ1 5 Properties of the Real Numbers The following properties can be used to manipulate algebraic expressions as well as real numbers Properties of the Real Numbers For any real numbers a, b, and c: a ϩ b b ϩ a and ab ba Commutative properties of addition and multiplication a ϩ ͑b ϩ c͒ ͑a ϩ b͒ ϩ c and a͑bc͒ ͑ab͒c Associative properties of addition and multiplication aϩ00ϩaa Additive identity property Ϫa ϩ a a ϩ ͑Ϫa͒ Additive inverse property aи11иaa Multiplicative identity property aи 1 и a ͑a a a a͑b ϩ c͒ ab ϩ ac 0͒ Multiplicative inverse property Distributive property Note that the distributive property is also true for subtraction since a͑b Ϫ c͒ a͓b ϩ ͑Ϫc͔͒ ab ϩ a͑Ϫc͒ ab Ϫ ac EXAMPLE State the property being illustrated in each sentence a) и и c) 14 ϩ ͑Ϫ14͒ e) 2͑a Ϫ b͒ 2a Ϫ 2b Solution SENTENCE b) ϩ ͑m ϩ n͒ ͑5 ϩ m͒ ϩ n d) и и PROPERTY a) и и b) c) d) e) Commutative property of multiplication: ab ba ϩ ͑m ϩ n͒ ͑5 ϩ m͒ ϩ n Associative property of addition: a ϩ ͑b ϩ c͒ ͑a ϩ b͒ ϩ c 14 ϩ ͑Ϫ14͒ Additive inverse property: a ϩ ͑Ϫa͒ 6и11и66 Multiplicative identity property: aи11иaa Distributive property: 2͑a Ϫ b͒ 2a Ϫ 2b a͑b ϩ c͒ ab ϩ ac Copyright © 2006 Pearson Education, Inc., publishing as Pearson Addison-Wesley BBEPMC0R_0312279093.QXP Chapter R 12/2/04 2:42 PM Page • Basic Concepts of Algebra Absolute Value The number line can be used to provide a geometric interpretation of absolute value The absolute value of a number a, denoted ͉a͉, is its distance from on the number line For example, ͉Ϫ5͉ 5, because the 3 distance of Ϫ5 from is Similarly, , because the distance of from is ԽԽ Absolute Value For any real number a, ͉a͉ ͭ a, if a Ն 0, Ϫa, if a Ͻ When a is nonnegative, the absolute value of a is a When a is negative, the absolute value of a is the opposite, or additive inverse, of a Thus, ͉a͉ is never negative; that is, for any real number a, ͉a͉ Ն Absolute value can be used to find the distance between two points on the number line a Distance Between Two Points on the Number Line For any real numbers a and b, the distance between a and b is ͉a Ϫ b͉, or equivalently, ͉b Ϫ a͉ b ͉a Ϫ b͉ ϭ ͉b Ϫ a͉ GCM EXAMPLE Solution Find the distance between Ϫ2 and The distance is ͉Ϫ2 Ϫ 3͉ ͉Ϫ5͉ 5, or equivalently, ͉3 Ϫ ͑Ϫ2͉͒ ͉3 ϩ 2͉ ͉5͉ We can also use the absolute-value operation on a graphing calculator to find the distance between two points On many graphing calculators, absolute value is denoted “abs” and is found in the MATH NUM menu and also in the CATALOG abs (Ϫ2Ϫ3) abs (3Ϫ(Ϫ2)) 5 Copyright © 2006 Pearson Education, Inc., publishing as Pearson Addison-Wesley BBEPMC0R_0312279093.QXP 12/2/04 2:42 PM Page Section R.1 R.1 The Real-Number System Exercise Set In Exercises 1– 10, consider the numbers Ϫ12, ͙7, 5.3, 25 ͓x, x ϩ h͔ [ Ϫ 73 , ͙ 8, 0, 5.242242224 , Ϫ͙14, ͙ 5, Ϫ1.96, 9, 23 , • ͙25, ͙ 4, Which are whole numbers? ͙ 8, 0, 9, ͙25 26 ͑x, x ϩ h͔ ( Ϫ12, ͙ 8, 0, 9, ͙25 ͙7, 5.242242224 , 3 Which are irrational numbers? Ϫ͙14, ͙ 5, ͙ 4 Which are natural numbers? ͙ 8, 9, ͙25 Ϫ12, 5.3, Ϫ , ͙ 8, 0, Which are rational numbers? Ϫ1.96, 9, , ͙25, 27 ͑ p, ϱ ͒ Which are real numbers? 28 ͑Ϫϱ, q͔ Which are integers? All of them 5.3, Ϫ , Ϫ1.96, Which are rational numbers but not integers? , Which are integers but not whole numbers? ] xϩh x ] xϩh x ( p ] q Ϫ12 13 ͕x ͉ Ϫ4 Յ x Ͻ Ϫ1͖ Ճ 14 ͕x ͉ Ͻ x Յ 6͖ Ճ In Exercises 29–46, the following notation is used: ގthe set of natural numbers, ޗthe set of whole numbers, ޚthe set of integers, ޑthe set of rational numbers, މthe set of irrational numbers, and ޒthe set of real numbers Classify the statement as true or false 29 ʦ ގTrue 30 ގTrue 15 ͕x ͉ x Յ Ϫ2͖ Ճ 16 ͕x ͉ x Ͼ Ϫ5͖ Ճ 18 ͕ x ͉ x Ն ͙3 ͖ Ճ 31 3.2 ʦ ޚ 17 ͕x ͉ x Ͼ 3.8͖ Ճ 19 ͕x ͉ Ͻ x͖ Ճ 20 ͕x ͉ Ϫ3 Ͼ x͖ Ճ Which are integers but not natural numbers? Ϫ12, 10 Which are real numbers but not integers? Ճ Write interval notation Then graph the interval 11 ͕x ͉ Ϫ3 Յ x Յ 3͖ Ճ 12 ͕x ͉ Ϫ4 Ͻ x Ͻ 4͖ Ճ 33 Ϫ Ϫ6 Ϫ5 Ϫ4 Ϫ3 Ϫ2 Ϫ1 ( ) 37 24 22 ͓Ϫ1, 2͔ [ Ϫ6 Ϫ5 Ϫ4 Ϫ3 Ϫ2 Ϫ1 ] 23 ͓Ϫ9, Ϫ4͒ [ Ϫ10 Ϫ9 Ϫ8 Ϫ7 Ϫ6 Ϫ5 ) Ϫ4 Ϫ3 Ϫ2 Ϫ1 ޒFalse ޗ 39 1.089 ] Ϫ10 Ϫ9 Ϫ8 Ϫ7 Ϫ6 Ϫ5 Ϫ4 Ϫ3 Ϫ2 Ϫ1 False މTrue 32 Ϫ10.1 ʦ ޒTrue 34 Ϫ͙6 ʦ ޑFalse 36 Ϫ1 ʦ ޗFalse 38 ʦ ޚ True 40 ގʕ ޗTrue 41 ޗʕ ޚTrue 42 ޚʕ ގFalse 43 ޑʕ ޒTrue 44 ޚʕ ޑTrue 45 ޒʕ ޚFalse 46 ޑʕ މFalse Name the property illustrated by the sentence 47 и x x и Commutative property of multiplication 48 ϩ ͑x ϩ y͒ ͑3 ϩ x͒ ϩ y of addition 24 ͑Ϫ9, Ϫ5͔ ( 11 ʦ ޑTrue 35 ͙11 Write interval notation for the graph 21 ͑0, 5͒ False Associative property 49 Ϫ3 и Ϫ3 50 x ϩ ϩ x Ճ Multiplicative identity property 51 5͑ab͒ ͑5a͒b Ճ 52 4͑ y Ϫ z͒ 4y Ϫ 4z Distributive property Ճ Answers to Exercises 10 – 20, 50, and 51 can be found on p IA-1 Copyright © 2006 Pearson Education, Inc., publishing as Pearson Addison-Wesley BBEPMC0R_0312279093.QXP Chapter R • 12/2/04 2:42 PM Page Basic Concepts of Algebra 53 2͑a ϩ b͒ ͑a ϩ b͒2 Additive inverse property 54 Ϫ7 ϩ not appear at the back of the book They are denoted by the words “Discussion and Writing.” 79 How would you convince a classmate that division is not associative? Multiplicative inverse property 80 Under what circumstances is ͙a a rational number? Commutative property of multiplication 55 Ϫ6͑m ϩ n͒ Ϫ6͑n ϩ m͒ Commutative property of addition 56 t ϩ t Additive identity property 57 и 58 9x ϩ 9y 9͑x ϩ y͒ Distributive property Synthesis Simplify 59 ͉Ϫ7.1͉ 7.1 60 ͉Ϫ86.2͉ 86.2 61 ͉347͉ 347 62 ͉Ϫ54͉ 54 To the student and the instructor: The Synthesis exercises found at the end of every exercise set challenge students to combine concepts or skills studied in that section or in preceding parts of the text Խ 63 Ϫ͙97 Խ ͙97 65 ͉0͉ 67 ͉͉ 64 ͉͉ 12 19 Between any two (different) real numbers there are many other real numbers Find each of the following Answers may vary 81 An irrational number between 0.124 and 0.125 12 19 66 ͉15͉ 15 Խ 68 Ϫ͙3 Խ ͙3 Find the distance between the given pair of points on the number line 69 Ϫ5, 11 70 Ϫ2.5, 2.5 15 23 , 12 24 71 Ϫ8, Ϫ2 72 73 6.7, 12.1 5.4 74 Ϫ14, Ϫ3 11 75 Ϫ 15 21 , 8 77 Ϫ7, Answers may vary; 0.124124412444 82 A rational number between Ϫ͙2.01 and Ϫ͙2 Answers may vary; Ϫ1.415 83 A rational number between Ϫ Answers may vary; Ϫ0.00999 1 and Ϫ 101 100 84 An irrational number between ͙5.99 and ͙6 Answers may vary; ͙5.995 85 The hypotenuse of an isosceles right triangle with legs of length unit can be used to “measure” a value for ͙2 by using the Pythagorean theorem, as shown 76 Ϫ3.4, 10.2 13.6 78 3, 19 16 c Collaborative Discussion and Writing To the student and the instructor: The Collaborative Discussion and Writing exercises are meant to be answered with one or more sentences These exercises can also be discussed and answered collaboratively by the entire class or by small groups Because of their open-ended nature, the answers to these exercises c 12 ϩ 12 c2 c ͙2 Draw a right triangle that could be used to “measure” ͙10 units Ճ Ճ Answer to Exercise 85 can be found on p IA-1 Copyright © 2006 Pearson Education, Inc., publishing as Pearson Addison-Wesley BBEPMC0R_0312279093.QXP 12/2/04 2:42 PM Page Section R.2 Simplify expressions with integer exponents Solve problems using scientific notation Use the rules for order of operations R.2 Integer Exponents, Scientific Notation, and Order of Operations • Integer Exponents, Scientific Notation, and Order of Operations Integers as Exponents When a positive integer is used as an exponent, it indicates the number of times a factor appears in a product For example, 73 means и и and 51 means For any positive integer n, a n a и a и a и и и a, n factors where a is the base and n is the exponent Zero and negative-integer exponents are defined as follows For any nonzero real number a and any integer m, a0 and aϪm EXAMPLE a) am Simplify each of the following b) ͑Ϫ3.4͒0 Solution a) 60 b) ͑Ϫ3.4͒0 EXAMPLE a) 4Ϫ5 Write each of the following with positive exponents b) ͑0.82͒Ϫ7 Solution a) 4Ϫ5 45 ͑0.82͒Ϫ͑Ϫ7͒ ͑0.82͒7 ͑0.82͒Ϫ7 x Ϫ3 1 y8 c) Ϫ8 x Ϫ3 и Ϫ8 и y y y x x b) Copyright © 2006 Pearson Education, Inc., publishing as Pearson Addison-Wesley c) x Ϫ3 y Ϫ8 BBEPMC0R_0312279093.QXP 10 Chapter R 12/2/04 2:42 PM Page 10 • Basic Concepts of Algebra The results in Example can be generalized as follows For any nonzero numbers a and b and any integers m and n, aϪm bn b Ϫn am (A factor can be moved to the other side of the fraction bar if the sign of the exponent is changed.) Write an equivalent expression without negative exponents: EXAMPLE x Ϫ3y Ϫ8 z Ϫ10 Solution Since each exponent is negative, we move each factor to the other side of the fraction bar and change the sign of each exponent: x Ϫ3y Ϫ8 z 10 z Ϫ10 x 3y The following properties of exponents can be used to simplify expressions Properties of Exponents For any real numbers a and b and any integers m and n, assuming is not raised to a nonpositive power: am и an amϩn Product rule m a amϪn ͑a an Quotient rule ͑am ͒n amn Power rule ͑ab͒ a b Raising a product to a power m ͩͪ a b m m m am ͑b bm a) y Ϫ5 и y Raising a quotient to a power 48x 12 16x d) ͑2s Ϫ2 ͒5 b) c) ͑t Ϫ3 ͒5 ͩ 0͒ Simplify each of the following EXAMPLE e) 0͒ ͪ 45x Ϫ4y 9z Ϫ8 Ϫ3 Copyright © 2006 Pearson Education, Inc., publishing as Pearson Addison-Wesley BBEPMN00_0321279115.QXP 1/7/05 3:54 PM Page A-71 Chapter y 21 yЈ xЈ 45Њ Ϫ4 Ϫ2 4 Ϫ4 Ϫ2 y Ϫ4 (yЈ)2 ϭ 6xЈ yЈ xЈ 30Њ Ϫ4 Ϫ2 y yЈ Ϫ4 y yЈ 4 (xЈ)2 x ϩ (yЈ)2 ϭ1 x yЈ yЈ ϭ Ϫ~(xЈ)2 xЈ 45Њ Ϫ4 x (yЈ)2 ϭ 8xЈ yЈ y 37 Ϫ8 Ϫ4 8.1Њ Ϫ8 Ϫ2 xЈ Ϫ4 Ϫ4 Ϫ2 63.4Њ Ϫ4 Ϫ4 y x (xЈ)2 (yЈ)2 ϩ ϭ1 16 48 Ϫ2 30Њ xЈ Ϫ2 29 Ϫ4 Ϫ2 (xЈ)2 (yЈ)2 Ϫ ϭ1 Ϫ4 Ϫ2 45Њ xЈ yЈ Ϫ2 27 xЈ 60Њ Ϫ4 Ϫ2 (yЈ ϩ 1)2 ϭ1 y 35 xЈ Ϫ Ϫ8 10 x Ϫ4 Ϫ4 25 (xЈ)2 Ϫ8 Ϫ4 x Ϫ2 y 33 yЈ 30Њ Ϫ2 (xЈ)2 (yЈ)2 Ϫ ϭ1 Ϫ4 xЈ x Ϫ2 23 y 31 yЈ A-71 x (xЈ)2 (yЈ)2 Ϫ ϭ1 54 46 39 Discussion and Writing 41 [5.3] 2 7 43 [5.3] 60° 44 [5.3] 135° 45 x xЈ cos Ϫ yЈ sin , y xЈ sin ϩ yЈ cos 47 AЈ ϩ CЈ A cos2 ϩ B sin cos ϩ C sin2 ϩ A sin2 Ϫ B sin cos ϩ C cos2 A͑sin2 ϩ cos2 ͒ ϩ C ͑sin2 ϩ cos2 ͒ AϩC 42 [5.3] Ϫ x Exercise Set 9.6 (b) (a) (d) (a) Parabola; (b) vertical, unit to the right of the pole; (c) ͑ 12 , ͒; BBEPMN00_0321279115.QXP A-72 (d) 1/7/05 3:54 PM Page A-72 Answers rϭ 17 (a) Parabola; (b) horizontal, 32 units below the pole; 3 ; , (c) (d) 1 ϩ cos u ͩ ͪ Ϫ9 rϭ Ϫ sin u Ϫ4 (a) Hyperbola; (b) horizontal, 32 units below the pole; 3 (c) Ϫ3, , 1, ; 2 (d) 15 ͩ ͪͩ ͪ rϭ Ϫ 10 sin u Ϫ6 Ϫ6 Ϫ3 19 (a) Ellipse; (b) vertical, units to the left of the pole; (c) ͑4, 0͒, ͑ 43 , ͒; (d) r ϭ Ϫ cos u Ϫ3 Ϫ6 11 (a) Ellipse; (b) vertical, 83 units to the left of the pole; (c) ͑ 83 , ͒, ͑ 89 , ͒; (d) rϭ Ϫ cos u Ϫ3 21 (a) Hyperbola; (b) horizontal, 107 units above the pole; 3 (c) , , Ϫ , ; 12 (d) ͩ ͪͩ rϭ Ϫ4 ͪ ϩ 10 sin u Ϫ3 13 (a) Hyperbola; (b) horizontal, 43 units above the pole; 3 (c) , , Ϫ4, ; 2 20 (d) ͩ ͪͩ ͪ rϭ 10 ϩ 15 sin u Ϫ9 Ϫ4 ϩ cos u Ϫ5 Ϫ3 Ϫ3 23 27 29 31 35 37 41 15 (a) Ellipse; (b) vertical, units to the right of the pole; (c) ͑1, 0͒, ͑3, ͒; (d) rϭ Ϫ6 45 49 51 52 53 54 55 25 x Ϫ 3y Ϫ 12y Ϫ y ϩ 2x Ϫ 2 27x ϩ 36y Ϫ 48x Ϫ 64 4x Ϫ 5y ϩ 24y Ϫ 16 33 4x Ϫ 12y Ϫ 3x ϩ 4y ϩ 6x Ϫ 2 3x ϩ 4y Ϫ 8x Ϫ 16 39 r 4x Ϫ 96y ϩ 140y Ϫ 49 ϩ sin 43 r r ϩ cos Ϫ cos 15 47 r r ϩ sin Ϫ sin Discussion and Writing [1.1] f ͑t͒ ͑t Ϫ 3͒2 ϩ 4, or t Ϫ 6t ϩ 13 [1.1] f ͑2t͒ ͑2t Ϫ 3͒2 ϩ 4, or 4t Ϫ 12t ϩ 13 [1.1] f ͑t Ϫ 1͒ ͑t Ϫ 4͒2 ϩ 4, or t Ϫ 8t ϩ 20 [1.1] f ͑t ϩ 2͒ ͑t Ϫ 1͒2 ϩ 4, or t Ϫ 2t ϩ 1.5 ϫ 108 r ϩ sin BBEPMN00_0321279115.QXP 1/7/05 3:54 PM Page A-73 Chapter Exercise Set 9.7 A-73 x ϭ e Ϫt, y ϭ e t; Ϫϱ Ͻ t Ͻ ϱ 11 10 1 x ϭ t, y ϭ 6t Ϫ 7; Ϫ1 р t р 40 Ϫ1 Ϫ5 10 Ϫ1 y Ϫ20 y 12x Ϫ 7, Ϫ 12 Յ x Յ 3 x ϭ 4t 2, y ϭ 2t; Ϫ1 р t р 1 ,x Ͼ x x ϭ cos t, y ϭ sin t; р t р 2π 13 Ϫ9 Ϫ2 Ϫ6 x ϩ y 9, Ϫ3 Յ x Յ x ϭ cos t, y ϭ sin t; р t р 2π 15 Ϫ5 x y , Ϫ2 Յ y Յ x ϭ t 2, y ϭ ͙ළt ; р t р 2 Ϫ6 Ϫ2 Ϫ4 25 Ϫ2 x y 4, Յ y Յ x ϭ t ϩ 3, y ϭ ; Ϫ2 р t р tϩ3 y2 x2 ϩ 1, Ϫ1 Յ x Յ π π 17 x ϭ sec t, y ϭ cos t; Ϫ Ͻ t Ͻ 2 2 Ϫ1 Ϫ1 10 10 Ϫ1 Ϫ1 y y ,x Ն x x ϭ ϩ cos t, y ϭ ϩ sin t; р t р 2π 19 ,1 Յ x Յ x x ϭ 2t Ϫ 1, y ϭ t 2; Ϫ3 р t р 15 Ϫ6 Ϫ10 10 Ϫ5 y ͑x ϩ 1͒2, Ϫ7 Յ x Յ Ϫ2 ͑x Ϫ 1͒ ϩ ͑ y Ϫ 2͒2 4, Ϫ1 Յ x Յ 21 0.7071 23 Ϫ0.2588 25 0.7265 27 0.5460 t 29 Answers may vary x t , y 4t Ϫ 3; x ϩ 3, ytϩ9 31 Answers may vary x t , y ͑t Ϫ 2͒2 Ϫ 6t ; x t ϩ 2, y t Ϫ 6t Ϫ 12 BBEPMN00_0321279115.QXP A-74 1/7/05 3:54 PM Page A-74 Answers [9.1] x Ϫ6y 10 [9.1] F: ͑Ϫ3, 0͒; V: ͑0, 0͒; 15 17 D: x 11 [9.1] V: ͑Ϫ5, 8͒; F: ͑ Ϫ5, ͒; D : y 12 [9.2] C: ͑2, Ϫ1͒; V: ͑Ϫ3, Ϫ1͒, ͑7, Ϫ1͒; F: ͑Ϫ1, Ϫ1͒, y ͑5, Ϫ1͒; 33 (a) x 40͙3t , y ϩ 40t Ϫ 16t 2; (b) x ϭ 40͙ළ3 t, y ϭ ϩ 40t Ϫ 16t (c) 31 ft, 23 ft; 50 16x ϩ 25y Ϫ 64x ϩ 50y Ϫ 311 ϭ 200 Ϫ2 Ϫ2 (d) about 2.7 sec; (e) about 187.1 ft; (f ) 32 ft 35 x ϭ 2(t Ϫ sin t), y ϭ 2(1 Ϫ cos t); р t р 4p x Ϫ4 x2 y2 ϩ 1 16 1 14 [9.3] C: Ϫ2, ; V: 0, ; , Ϫ4, 4 13 [9.2] 30 ͩ 1 , Ϫ2 Ϫ ͙6, ; 4 1 ͙2 ͙2 A: y Ϫ ͑x ϩ 2͒, y Ϫ Ϫ ͑x ϩ 2͒ 4 15 [9.1] 0.167 ft 16 [9.4] ͑ Ϫ8͙2, ͒, ͑ 8͙2, ͒ x ϭ t Ϫ sin t, y ϭ Ϫ cos t ; Ϫ2p р t р 2p 37 F: Ϫ2 ϩ ͙6, Ϫ10 10 39 Discussion and Writing 41 [1.1] y 42 [1.1] yϭ x ϭ y3 Ϫ4 Ϫ2 x x Ϫ2 Ϫ4 43 [1.2] 44 [3.5] y y 19 21 22 23 24 25 26 29 31 f(x) ϭ ͙x Ϫ 2 Ϫ2 Ϫ2 Ϫ4 Ϫ2 x Ϫ5 Ϫ4 Ϫ3 x Ϫ2 f (x) ϭ Ϫ4 x2 45 x cos t , y Ϫ3 sin t Review Exercises: Chapter [9.1] (d) [9.2] (b) [9.2] (a) [9.2] (f ) [9.2] (e) [9.1] (h) [9.3] (g) [9.3] (c) Ϫ1 ͪͩ ͪͩ ͪ ͙29 ͙29 ͙29 , Ϫ3, , 3, Ϫ , 2 ͪ ͙29 18 [9.4] ͑7, 4͒ 32 [9.4] ͑2, 2͒, ͑ , Ϫ 109 ͒ 20 [9.4] ͑0, Ϫ3͒, ͑2, 1͒ [9.4] ͑4, 3͒, ͑4, Ϫ3͒, ͑Ϫ4, 3͒, ͑Ϫ4, Ϫ3͒ [9.4] ͑ Ϫ͙3, ͒, ͑ ͙3, ͒, ͑Ϫ2, 1͒, ͑2, 1͒ [9.4] ͑ Ϫ 35 , 215 ͒, ͑3, Ϫ3͒ [9.4] ͑6, 8͒, ͑6, Ϫ8͒, ͑Ϫ6, 8͒, ͑Ϫ6, Ϫ8͒ [9.4] ͑2, 2͒, ͑Ϫ2, Ϫ2͒, ͑ 2͙2, ͙2 ͒, ͑ Ϫ2͙2, Ϫ͙2 ͒ [9.4] 7, 27 [9.4] m by 12 m 28 [9.4] 4, [9.4] 32 cm, 20 cm 30 [9.4] 11 ft, ft [9.4] 32 [9.4] y y Ϫ3, Ϫ x3 Ϫ2 ͩ y Ϫ4 Ϫ2 ͩ 17 [9.4] 3, Ϫ3 ͩ ͪ ͩ ͪͩ ͪ ͪͩ ͪ (Ϫ3, Ϫ5) 5 3 Ϫ1 Ϫ1 Ϫ2 Ϫ3 Ϫ4 Ϫ5 (2, 0) x Ϫ5 Ϫ3 Ϫ2 Ϫ1 Ϫ1 Ϫ2 Ϫ3 Ϫ5 (0, 4) (4, 0) x BBEPMN00_0321279115.QXP 1/7/05 3:54 PM Page A-75 Chapter 33 [9.4] 34 [9.4] y (Ϫ͙2, 1) Ϫ Ϫ Ϫ Ϫ2 Ϫ1 Ϫ5 Ϫ4 x Ϫ2 Ϫ3 Ϫ4 Ϫ5 Ϫ2 Ϫ1 Ϫ1 Ϫ2 (Ϫ1, Ϫ2͙2) Ϫ4 Ϫ5 y 4 x ͩ ͪ Horizontal directrix units below the pole; vertex: 1, rϭ ϩ cos u Ϫ6 12 Ϫ6 Ϫ cos u 45Њ Ϫ4 Ϫ2 Vertical directrix units to the right of the pole; vertices: ͑ 0, 43 ͒, ͑Ϫ4, ͒ 41 [9.6] r ϭ xЈ Ϫ2 6 y Ϫ4 Ϫ2 Ϫ6 x (xЈ)2 (yЈ)2 ϩ ϭ1 Ϫ4 yЈ 45Њ Ϫ2 36 [9.5] xЈ Ϫ4 Ϫ2 40 [9.6] yЈ Ϫ sin u (͙2, 1) 35 [9.5] rϭ (Ϫ1, 2͙2) 39 [9.6] y (xЈ)2 Ϫ x (yЈ)2 ϭ1 Ϫ3 Ϫ3 37 [9.5] y yЈ (xЈ)2 (yЈ)2 Ϫ ϭ1 Vertical directrix units to the left of the pole; vertices: ͑4, 0͒, ͑ 43 , ͒ 18 42 [9.6] rϭ xЈ Ϫ4 Ϫ2 30Њ ϩ sin u x Ϫ2 Ϫ9 Ϫ4 38 [9.5] y yЈ (xЈ)2 ϭ ϪyЈ xЈ Ϫ4 Ϫ2 45Њ Ϫ2 Ϫ4 A-75 x Ϫ8 ͩ ͪͩ ͪ Horizontal directrix units above the pole; 3 vertices: , , 6, 2 43 [9.6] x Ϫ 4y Ϫ 44 [9.6] 3x Ϫ y Ϫ 16x ϩ 16 45 [9.6] 3x ϩ 4y Ϫ 8x Ϫ 16 46 [9.6] 9x ϩ 5y ϩ 24y Ϫ 36 47 [9.6] r , or r ϩ cos ϩ 12 cos 18 48 [9.6] r 49 [9.6] r Ϫ sin Ϫ cos 50 [9.6] r ϩ sin 3 BBEPMN00_0321279115.QXP A-76 1/7/05 3:54 PM Page A-76 Answers x ϭ t, y ϭ ϩ t; Ϫ3 р t р 51 [9.7] 59 Discussion and Writing [9.2] The equation of a circle can be written as ͑x Ϫ h͒2 ͑ y Ϫ k͒2 ϩ 1, a2 b2 Ϫ3 Ϫ2 y ϩ x, Ϫ3 Յ x Յ 52 [9.7] x ϭ ͙t, y ϭ t Ϫ 1; р t р 10 Ϫ1 where a b r, the radius of the circle In an ellipse, a Ͼ b, so a circle is not a special type of ellipse y2 60 [9.2] x ϩ 61 [9.4] , 62 [9.2], [9.4] ͑x Ϫ 2͒2 ϩ ͑ y Ϫ 1͒2 100 x2 y2 63 [9.3] Ϫ 1 778.41 39,221.59 Test: Chapter Ϫ2 y x Ϫ 1, Յ x Յ x ϭ cos t, y ϭ sin t; р t р 2p 53 [9.7] [9.3] (c) [9.1] (b) [9.2] (a) [9.1] V: ͑0, 0͒; F: ͑0, 3͒; D: y Ϫ3 y Ϫ4.5 4.5 Ϫ4 Ϫ2 x Ϫ2 Ϫ3 x2 ϩ y2 54 [9.7] x ϭ sin t, y ϭ cos t; р t р 2p Ϫ4 x ϭ 12y [9.1] V: ͑Ϫ1, Ϫ1͒; F: ͑1, Ϫ1͒; D : x Ϫ3 y Ϫ4.5 4.5 Ϫ3 x2 ϩ y2 55 [9.7] Answers may vary x t , y 2t Ϫ 3; x t ϩ 1, y 2t Ϫ 56 [9.7] Answers may vary x t , y t ϩ 4; x t Ϫ 2, y t Ϫ 4t ϩ 57 [9.7] (a) x 75͙2t , y 75͙2t Ϫ 16t 2; (b) x ϭ 75͙ළ2 t, y ϭ 75͙ළ2 t Ϫ 16t (c) 174.2 ft, 60.4 ft; Ϫ4 Ϫ2 x Ϫ2 Ϫ4 y ϩ 2y Ϫ 8x Ϫ ϭ [9.1] x 8y [9.2] Center: ͑Ϫ1, 3͒; radius: y 250 0 800 (d) about 6.6 sec; (e) about 700.0 ft; (f ) about 175.8 ft 58 Discussion and Writing [9.4] Although we can always visualize the real-number solutions, we cannot visualize the imaginary-number solutions Ϫ6 Ϫ4 Ϫ2 x Ϫ2 x ϩ y ϩ 2x Ϫ 6y Ϫ 15 ϭ [9.2] (d) BBEPMN00_0321279115.QXP 1/7/05 3:54 PM Page A-77 A-77 Chapter 9 [9.2] C: ͑0, 0͒; V: ͑Ϫ4, 0͒, ͑4, 0͒; F: ͑ Ϫ͙7, ͒, ͑ ͙7, ͒ y 16 17 19 21 Ϫ4 Ϫ2 x 27 ͙2 ͙2 15 [9.1] in x, y Ϫ x 2 [9.4] ͑1, 2͒, ͑1, Ϫ2͒, ͑Ϫ1, 2͒, ͑Ϫ1, Ϫ2͒ [9.4] ͑3, Ϫ2͒, ͑Ϫ2, 3͒ 18 [9.4] ͑2, 3͒, ͑3, 2͒ [9.4] ft by ft 20 [9.4] 60 ft by 45 ft y [9.4] 14 [9.3] y (3, 5) Ϫ2 Ϫ4 Ϫ5 Ϫ4 Ϫ3 9x ϩ 16y ϭ 144 Ϫ1 Ϫ1 x (Ϫ1, Ϫ3) Ϫ3 10 [9.2] C: ͑Ϫ1, 2͒; V: ͑Ϫ1, Ϫ1͒, ͑Ϫ1, 5͒; F: ͑ Ϫ1, Ϫ ͙5 ͒, ͑ Ϫ1, ϩ ͙5 ͒ Ϫ5 y 22 [9.5] After using the rotation of axes formulas with ͑xЈ͒2 45Њ, we have ϩ ͑ yЈ͒2 y Ϫ4 Ϫ2 x y' Ϫ2 x' Ϫ4 Ϫ4 45Њ Ϫ2 (x ϩ 1) (y Ϫ 2) ϪϪϪ4ϪϪ ϩ ϪϪϪ 9ϪϪ ϭ x Ϫ2 x2 y2 11 [9.2] ϩ 1 25 12 [9.3] C: ͑0, 0͒; V: ͑Ϫ1, 0͒, ͑1, 0͒; F: ͑ Ϫ͙5, ͒, ͑ ͙5, ͒; y A: y Ϫ2x , y 2x Ϫ4 23 [9.6] rϭ Ϫ sin Ϫ3 Ϫ4 Ϫ2 x Ϫ2 Ϫ2 Ϫ4 4x Ϫ y ϭ 13 [9.3] C: ͑Ϫ1, 2͒; V: ͑Ϫ1, 0͒, ͑Ϫ1, 4͒; F: ͑ Ϫ1, Ϫ ͙13 ͒, ͑ Ϫ1, ϩ ͙13 ͒; A: y Ϫ 23 x ϩ 43 , y 23 x ϩ 83 ͩ ͪ Horizontal directrix units below the pole; vertex: 1, 24 [9.6] r ϩ cos 3 x ϭ ͙ළt , y ϭ t ϩ 2; р t р 16 25 [9.7] 20 y Ϫ1 Ϫ4 Ϫ2 Ϫ1 26 [9.7] x ϩ y 9, Ϫ3 Յ x Յ 27 [9.7] Answer may vary x t , y t Ϫ 5; x t ϩ 5, yt Ϫ2 Ϫ4 (y Ϫ 2) x (x ϩ 1) ϪϪϪ4ϪϪ Ϫ ϪϪϪ 9ϪϪ ϭ BBEPMN00_0321279115.QXP A-78 1/7/05 3:54 PM Page A-78 Answers 28 [9.7] (a) x 125͙3t , y 10 ϩ 125t Ϫ 16t 2; (b) x ϭ 125͙ළ3t, y ϭ 10 ϩ 125t Ϫ 16t (c) 119 ft, 241 ft; 300 2000 (d) about 7.9 sec; (e) about 1710.4 ft; (f ) about 254.1 ft 29 [9.2] ͑x Ϫ 3͒2 ϩ ͑ y ϩ 1͒2 Chapter 10 Exercise Set 10.1 10 3, 7, 11, 15; 39; 59 2, , , ; ; 15 14 99 112 0, 35 , 54 , 15 Ϫ1, 4, Ϫ9, 16; 100; Ϫ225 17 ; 101 ; 113 7, 3, 7, 3; 3; 11 34 13 225 15 Ϫ33,880 17 67 19 n Un 21 10 2.25 2.3704 2.4414 2.4883 2.5216 2.5465 2.5658 2.5812 2.5937 n Un 10 1.5538 1.4988 1.4914 1.4904 1.4902 1.4902 1.4902 1.4902 1.4902 10 ͩ ͪ Exercise Set 10.2 10 nϩ1 nϩ2 29 n͑n ϩ 1͒ 31 log 10nϪ1, or n Ϫ 33 6; 28 35 20; 30 37 12 ϩ 14 ϩ 16 ϩ 18 ϩ 101 137 120 39 ϩ ϩ ϩ ϩ 16 ϩ 32 ϩ 64 127 23 2n 41 ln ϩ ln ϩ ln ϩ ln 10 ln ͑7 и и и 10͒ ln 5040 Ϸ 8.5252 43 12 ϩ 23 ϩ 34 ϩ 45 ϩ 56 ϩ 67 ϩ 78 ϩ 89 15,551 2520 45 Ϫ1 ϩ Ϫ ϩ Ϫ Ϫ1 47 Ϫ ϩ Ϫ 12 ϩ 15 Ϫ 18 ϩ 21 Ϫ 24 Ϫ12 49 ϩ ϩ 25 ϩ 15 ϩ 172 ϩ 131 ϩ 372 157,351 40,885 51 ϩ ϩ ϩ ϩ 11 ϩ 18 43 32 64 128 256 512 53 12 ϩ 23 ϩ 45 ϩ 89 ϩ 16 17 ϩ 33 ϩ 65 ϩ 129 ϩ 257 ϩ 513 ϩ 1024 1025 Ϸ 9.736 ϱ 6 k 55 57 59 5k ͑Ϫ1͒kϩ12k ͑Ϫ1͒k kϩ1 k1 k1 k1 n ϱ 1 k 61 63 65 4, 4, 45, 59 ͑Ϫ1͒ k k1 k͑k ϩ 1͒ k2 67 6561, Ϫ81, 9i, Ϫ3͙i 69 2, 3, 5, 71 (a) 1062, 1127.84, 1197.77, 1272.03, 1350.90, 1434.65, 1523.60, 1618.07, 1718.39, 1824.93; (b) $3330.35 73 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16,384, 32,768, 65,536 75 1, 1, 2, 3, 5, 8, 13 77 (a) a n Ϫ13.21231156n ϩ 52,871.41834n Ϫ 52,893,240.07; (b) 253 thousand, 350 thousand, 342 thousand, 131 thousand 79 Discussion and Writing 80 [8.1], [8.3], [8.5], [8.6] ͑Ϫ1, Ϫ3͒ 81 [8.1], [8.3], [8.5], [8.6] 2002: 569,700,000 listings; 2003: 971,000,000 listings 82 [9.2] ͑3, Ϫ2͒; ͙97 83 [9.2] Ϫ , ; 85 i, Ϫ1, Ϫi, 1, i; i 2 87 ln ͑1 и и и и и и и n͒ 25 ͑Ϫ1͒n и и 3nϪ1 27 a1 3, d a 9, d Ϫ4 a 32 , d 34 a $316, d Ϫ$3 a 12 46 11 a 14 Ϫ 173 13 a 10 $7941.62 15 33rd 17 46th 19 a 21 n 39 23 a 13 ; d 12 ; 13 , 65 , 34 , 116 , 73 25 670 27 160,400 29 735 31 990 33 1760 35 652 37 Ϫ 6026 39 1260 poles 41 4960¢, or $49.60 13 43 1320 seats 45 plants; 171 plants 47 Yes; 49 Discussion and Writing 50 [8.1], [8.3], [8.5], [8.6] ͑2, 5͒ 51 [8.2], [8.3], [8.5], [8.6] ͑2, Ϫ1, 3͒ 52 [9.2] ͑Ϫ4, 0͒, ͑4, 0͒; ͑ Ϫ͙7, ͒, ͑ ͙7, ͒ x2 y2 53 [9.2] 55 n ϩ 1 25 57 a 60 Ϫ 5p Ϫ 5q ; d 5p ϩ 2q Ϫ 20 59 6, 8, 10 61 45 , 35 , 25 , 11 15 63 Insert 16 arithmetic means between and 50 with d 49 17 65 m p ϩ d mqϪd 2m p ϩ q pϩq m Adding BBEPMN00_0321279115.QXP 1/7/05 3:54 PM Page A-79 Chapters – 10 Visualizing the Graph J D A B C G 10 I F H E Exercise Set 10.3 Ϫ1 Ϫ2 0.1 a 11 128 19 ͑Ϫ1͒nϪ1 Exercise Set 10.4 12 Ͻ 13, false; 22 Ͻ 23, true; 32 Ͻ 33, true; 42 Ͻ 43, true; 52 Ͻ 53, true 3͑3 Ϫ 3͒ A polygon of sides has diagonals True; A 4͑4 Ϫ 3͒ polygon of sides has diagonals True; A polygon 5͑5 Ϫ 3͒ of sides has diagonals True; A polygon of 6͑6 Ϫ 3͒ sides has diagonals True; A polygon of sides 7͑7 Ϫ 3͒ has diagonals True S n : ϩ ϩ ϩ и и и ϩ 2n n͑n ϩ 1͒ S 1: 1͑1 ϩ 1͒ Sk : ϩ ϩ ϩ и и и ϩ 2k k͑k ϩ 1͒ S kϩ1: ϩ ϩ ϩ и и и ϩ 2k ϩ 2͑k ϩ 1͒ ͑k ϩ 1͒ ͑k ϩ 2͒ (1) Basis step: S true by substitution (2) Induction step: Assume S k Deduce S kϩ1 Starting with the left side of S kϩ1, we have ϩ ϩ ϩ и и и ϩ 2k ϩ 2͑k ϩ 1͒ By Sk k͑k ϩ 1͒ ϩ 2͑k ϩ 1͒ Distributive law ͑k ϩ 1͒ ͑k ϩ 2͒ S n : ϩ ϩ ϩ и и и ϩ ͑4n Ϫ 3͒ n͑2n Ϫ 1͒ S 1: 1͑2 и Ϫ 1͒ Sk : ϩ ϩ ϩ и и и ϩ ͑4k Ϫ 3͒ k͑2k Ϫ 1͒ S kϩ1: ϩ ϩ ϩ и и и ϩ ͑4k Ϫ 3͒ ϩ ͓4͑k ϩ 1͒ Ϫ 3͔ ͑k ϩ 1͒ ͓2͑k ϩ 1͒ Ϫ 1͔ ͑k ϩ 1͒ ͑2k ϩ 1͒ (1) Basis step: S true by substitution (2) Induction step: Assume S k Deduce S kϩ1 Starting with the left side of S kϩ1, we have ϩ ϩ ϩ и и и ϩ ͑4k Ϫ 3͒ ϩ ͓4͑k ϩ 1͒ Ϫ 3͔ By Sk k͑2k Ϫ 1͒ ϩ ͓4͑k ϩ 1͒ Ϫ 3͔ 2k Ϫ k ϩ 4k ϩ Ϫ ͑k ϩ 1͒ ͑2k ϩ 1͒ S n : ϩ ϩ ϩ и и и ϩ 2n 2͑2n Ϫ 1͒ S 1: 2͑2 Ϫ 1͒ Sk : ϩ ϩ ϩ и и и ϩ 2k 2͑2k Ϫ 1͒ S kϩ1: ϩ ϩ ϩ и и и ϩ 2k ϩ 2kϩ1 2͑2kϩ1 Ϫ 1͒ (1) Basis step: S is true by substitution (2) Induction step: Assume S k Deduce S kϩ1 Starting with the left side of S kϩ1, we have ϩ ϩ ϩ и и и ϩ 2k ϩ 2kϩ1 13 162 15 7͑5͒40 17 3nϪ1 21 n 23 762 25 4921 27 True 29 True 18 x 31 True 33 35 125 37 Does not exist 39 32 38,569 41 29 59,049 43 45 Does not exist 47 $4545.45 13 34,091 49 160 51 53 55 57 (a) 99 9990 256 ft; (b) ft 59 (a) About 297 ft; (b) 300 ft 61 $23,841.50 63 $523,619.17 65 $86,666,666,667 67 Discussion and Writing 69 [1.6] ͑ f ؠg͒ ͑x͒ 16x ϩ 40x ϩ 25; 70 [1.6] ͑ f ؠg͒ ͑x͒ x ϩ x ϩ 2; ͑ g ؠf ͒ ͑x͒ 4x ϩ 71 [4.5] 2.209 72 [4.5] 161 ͑ g ؠf ͒ ͑x͒ x Ϫ x ϩ 73 ͑ Ϫ ͙6 ͒͑͞ ͙3 Ϫ ͙2 ͒ 2͙3 ϩ ͙2, ͑ 6͙3 Ϫ 2͙2 ͒͑͞ Ϫ ͙6 ͒ 2͙3 ϩ ͙2; there exists a common ratio, 2͙3 ϩ ͙2; thus the sequence is geometric 75 (a) 133 ; 223 , 343 , 463 , 583 ; (b) Ϫ 113 ; Ϫ 23 , 103 , Ϫ 503 , 250 or 5; 8, 12, x 2͑1 Ϫ ͑Ϫx͒n ͒ 18, 27 77 S n xϩ1 a nϩ1 a nϩ1 a nϩ1 79 r , so ln ln r But ln ln a nϩ1 Ϫ an an an ln a n ln r Thus, ln a 1, ln a 2, is an arithmetic sequence with common difference ln r 81 512 cm2 A-79 By Sk 2͑2k Ϫ 1͒ ϩ 2kϩ1 kϩ1 kϩ1 Ϫ2ϩ2 2 и 2kϩ1 Ϫ 2͑2kϩ1 Ϫ 1͒ 11 (1) Basis step: Since Ͻ ϩ 1, S is true (2) Induction step: Assume S k Deduce S kϩ1 Now By Sk kϽkϩ1 Adding kϩ1Ͻkϩ1ϩ1 k ϩ Ͻ k ϩ 13 (1) Basis step: Since 2, S is true (2) Induction step: Let k be any natural number Assume S k Deduce S kϩ1 By Sk 2k Յ 2k Multiplying by 2 и 2k Յ и 2k 4k Յ 2kϩ1 Since Յ k , k ϩ Յ k ϩ k , or k ϩ Յ 2k Then 2͑k ϩ 1͒ Յ 4k Thus, 2͑k ϩ 1͒ Յ 4k Յ 2kϩ1, so 2͑k ϩ 1͒ Յ 2kϩ1 BBEPMN00_0321279115.QXP A-80 1/7/05 3:54 PM Page A-80 Answers 15 k͑k ϩ 3͒ ϩ 4͑k ϩ 1͒ ͑k ϩ 2͒ ͑k ϩ 1͒ ͑k ϩ 2͒ ͑k ϩ 3͒ k͑k ϩ 3͒͑k ϩ 3͒ ϩ 4͑k ϩ 1͒ ͑k ϩ 2͒ ͑k ϩ 3͒ k ϩ 6k ϩ 9k ϩ 4͑k ϩ 1͒ ͑k ϩ 2͒ ͑k ϩ 3͒ ͑k ϩ 1͒2͑k ϩ 4͒ 4͑k ϩ 1͒ ͑k ϩ 2͒ ͑k ϩ 3͒ ͑k ϩ 1͒ ͑k ϩ 4͒ 4͑k ϩ 2͒ ͑k ϩ 3͒ 17 Sn : ϩ ϩ ϩ иии ϩ n S 1: 1 Sk : ϩ ϩ ϩ иии ϩ k n͑n ϩ 1͒ 1͑1 ϩ 1͒ k͑k ϩ 1͒ S kϩ1: ϩ ϩ ϩ и и и ϩ k ϩ ͑k ϩ 1͒ (1) Basis step: S true by substitution ͑k ϩ 1͒ ͑k ϩ 2͒ (2) Induction step: Assume S k Deduce S kϩ1 Starting with the left side of S kϩ1, we have ϩ ϩ ϩ и и и ϩ k ϩ ͑k ϩ 1͒ 1 Sn : ϩ ϩ ϩ иии 1и2и3 2и3и4 3и4и5 n͑n ϩ 3͒ ϩ n͑n ϩ 1͒ ͑n ϩ 2͒ 4͑n ϩ 1͒ ͑n ϩ 2͒ 1͑1 ϩ 3͒ S 1: и и 4͑1 ϩ 1͒ ͑1 ϩ 2͒ 1 Sk : ϩ ϩ иии ϩ 1и2и3 2и3и4 k͑k ϩ 1͒ ͑k ϩ 2͒ k͑k ϩ 3͒ 4͑k ϩ 1͒ ͑k ϩ 2͒ 1 S kϩ1: ϩ ϩ иии ϩ 1и2и3 2и3и4 k͑k ϩ 1͒ ͑k ϩ 2͒ ϩ ͑k ϩ 1͒ ͑k ϩ 2͒ ͑k ϩ 3͒ ͑k ϩ 1͒ ͑k ϩ 4͒ ͑k ϩ 1͒ ͑k ϩ ϩ 3͒ 4͑k ϩ ϩ 1͒ ͑k ϩ ϩ 2͒ 4͑k ϩ 2͒ ͑k ϩ 3͒ 1 1͑1 ϩ 3͒ (1) Basis step: Since and 1и2и3 4͑1 ϩ 1͒ ͑1 ϩ 2͒ 1и4 , S is true 4и2и3 (2) Induction step: Assume S k Deduce S kϩ1 Add on both sides of S k and ͑k ϩ 1͒ ͑k ϩ 2͒ ͑k ϩ 3͒ simplify the right side Only the right side is shown here k͑k ϩ 1͒ By Sk ϩ ͑k ϩ 1͒ k͑k ϩ 1͒ ϩ 2͑k ϩ 1͒ Adding ͑k ϩ 1͒ ͑k ϩ 2͒ Distributive law 2 ͑1 ϩ 1͒ 19 (1) Basis step S 1: 13 True (2) Induction step: Assume S k Deduce S kϩ1 k 2͑k ϩ 1͒2 S k : 13 ϩ 23 ϩ и и и ϩ k k 2͑k ϩ 1͒2 BySk ϩ ͑k ϩ 1͒3 ͑k ϩ 1͒ ͓k ϩ 4͑k ϩ 1͔͒ ͑k ϩ 1͒ ͑k ϩ 2͒2 13 ϩ 23 ϩ и и и ϩ ͑k ϩ 1͒3 21 (1) Basis step S 1: 15 12͑1 ϩ 1͒2͑2 и 12 ϩ и Ϫ 1͒ 12 True (2) Induction step Assume S k: k 2͑k ϩ 1͒2͑2k ϩ 2k Ϫ 1͒ 12 Then 15 ϩ 25 ϩ и и и ϩ k ϩ ͑k ϩ 1͒5 k 2͑k ϩ 1͒2͑2k ϩ 2k Ϫ 1͒ ϩ ͑k ϩ 1͒5 12 k 2͑k ϩ 1͒2͑2k ϩ 2k Ϫ 1͒ ϩ 12͑k ϩ 1͒5 12 ͑k ϩ 1͒2͑2k ϩ 14k ϩ 35k ϩ 36k ϩ 12͒ 12 ͑k ϩ 1͒2͑k ϩ 2͒2͑2k ϩ 6k ϩ 3͒ 12 ͑k ϩ 1͒2͑k ϩ ϩ 1͒2͑2͑k ϩ 1͒2 ϩ 2͑k ϩ 1͒ Ϫ 1͒ 12 23 1͑1 ϩ 1͒ ͑1 ϩ 2͒ (1) Basis step S 1: 1͑1 ϩ 1͒ True (2) Induction step Assume S k : 1͑1 ϩ 1͒ ϩ 2͑2 ϩ 1͒ ϩ и и и ϩ k͑k ϩ 1͒ k͑k ϩ 1͒ ͑k ϩ 2͒ 15 ϩ 25 ϩ и и и ϩ k BBEPMN00_0321279115.QXP 1/7/05 3:54 PM Page A-81 Chapter 10 Then 1͑1 ϩ 1͒ ϩ 2͑2 ϩ 1͒ ϩ и и и ϩ k͑k ϩ 1͒ ϩ ͑k ϩ 1͒ ͑k ϩ ϩ 1͒ k͑k ϩ 1͒ ͑k ϩ 2͒ ϩ ͑k ϩ 1͒ ͑k ϩ 2͒ ͑k ϩ 1͒ ͑k ϩ 2͒ ͑k ϩ 3͒ ͑k ϩ 1͒ ͑k ϩ ϩ 1͒ ͑k ϩ ϩ 2͒ 25 (1) Basis step: Since 12 ͓2a ϩ ͑1 Ϫ 1͒d ͔ 12 и 2a a 1, S is true (2) Induction step: Assume S k Deduce S kϩ1 Starting with the left side of S kϩ1, we have a ϩ ͑a ϩ d͒ ϩ и и и ϩ ͓a ϩ ͑k Ϫ 1͒d ͔ ϩ ͓a ϩ kd ͔ k ͓2a ϩ ͑k Ϫ 1͒d ͔ ϩ ͓a ϩ kd ͔ By Sk k͓2a ϩ ͑k Ϫ 1͒d ͔ 2͓a ϩ kd ͔ ϩ 2 2ka ϩ k͑k Ϫ 1͒d ϩ 2a ϩ 2kd 2a 1͑k ϩ 1͒ ϩ k͑k Ϫ 1͒d ϩ 2kd 2a 1͑k ϩ 1͒ ϩ ͑k Ϫ ϩ 2͒kd 2a 1͑k ϩ 1͒ ϩ ͑k ϩ 1͒kd kϩ1 ͓2a ϩ kd ͔ 27 Discussion and Writing 28 [8.1], [8.3], [8.5], [8.6] ͑5, 3͒ 29 [8.2], [8.3], [8.5], [8.6] ͑2, Ϫ3, 4͒ 30 [8.1], [8.3], [8.5], [8.6] Hardback: 50; paperback: 30 31 [8.2], [8.3], [8.5], [8.6] $800 at 1.5%, $1600 at 2%, $2000 at 3% 33 (1) Basis step S 1: x ϩ y is a factor of x Ϫ y True S 2: x ϩ y is a factor of x Ϫ y True (2) Induction step Assume S kϪ1: x ϩ y is a factor of x 2͑kϪ1͒ Ϫ y 2͑kϪ1͒ Then x 2͑kϪ1͒ Ϫ y 2͑kϪ1͒ ͑x ϩ y͒Q͑x͒ for some polynomial Q Assume S k : x ϩ y is a factor of x 2k Ϫ y 2k Then x 2k Ϫ y 2k ͑x ϩ y͒P͑x͒ for some polynomial P x 2͑kϩ1͒ Ϫ y 2͑kϩ1͒ ͑x 2k Ϫ y 2k ͒ ͑x ϩ y ͒ Ϫ ͑x 2͑kϪ1͒ Ϫ y 2͑kϪ1͒ ͒ ͑x 2y ͒ ͑x ϩ y͒P͑x͒ ͑x ϩ y ͒ Ϫ ͑x ϩ y͒Q͑x͒ ͑x 2y ͒ ͑x ϩ y͒ ͓P͑x͒ ͑x ϩ y ͒ Ϫ Q͑x͒ ͑x 2y ͔͒ so x ϩ y is a factor of x 2͑kϩ1͒ Ϫ y 2͑kϩ1͒ A-81 35 S 2: Sk : loga ͑b1b ͒ loga b1 ϩ loga b2 loga ͑b1b и и и b k ͒ loga b1 ϩ loga b2 ϩ и и и ϩ loga bk S kϩ1: loga ͑b1b и и и b kϩ1 ͒ loga b1 ϩ loga b2 ϩ и и и ϩ loga bkϩ1 (1) Basis step: S is true by the properties of logarithms (2) Induction step: Let k be a natural number k Ն Assume S k Deduce S kϩ1 Left side of Sk؉1 loga ͑b1b и и и b kϩ1 ͒ By S loga ͑b1b и и и b k ͒ ϩ loga bkϩ1 loga b1 ϩ loga b ϩ и и и ϩ loga b k ϩ loga bkϩ1 37 S 2: z1 ϩ z2 z1 ϩ z2: ͑a ϩ bi͒ ϩ ͑c ϩ di͒ ͑a ϩ c͒ ϩ ͑b ϩ d͒i ͑a ϩ c͒ Ϫ ͑b ϩ d͒i ͑a ϩ bi͒ ϩ ͑c ϩ di͒ a Ϫ bi ϩ c Ϫ di ͑a ϩ c͒ Ϫ ͑b ϩ d͒i S k : z1 ϩ z2 ϩ и и и ϩ zk z1 ϩ z2 ϩ и и и ϩ zk ͑z1 ϩ z2 ϩ иии ϩ zk ͒ ϩ zkϩ1 ͑z1 ϩ z2 ϩ иии ϩ zk ͒ ϩ zkϩ1 By S By Sk z1 ϩ z2 ϩ и и и ϩ zk ϩ zkϩ1 39 S 1: i is either i or Ϫ1 or Ϫi or S k : i k is either i or Ϫ1 or Ϫi or i kϩ1 i k и i is then i и i Ϫ1 or Ϫ1 и i Ϫi or Ϫi и i or и i i 41 S 1: is a factor of 13 ϩ и is a factor of k ϩ 2k , i.e., k ϩ 2k и m Sk : S kϩ1: is a factor of ͑k ϩ 1͒3 ϩ 2͑k ϩ 1͒ Consider ͑k ϩ 1͒3 ϩ 2͑k ϩ 1͒ k ϩ 3k ϩ 5k ϩ ͑k ϩ 2k͒ ϩ 3k ϩ 3k ϩ 3m ϩ 3͑k ϩ k ϩ 1͒ A multiple of Exercise Set 10.5 720 604,800 120 3024 11 120 13 120 15 17 6,497,400 19 n͑n Ϫ 1͒ ͑n Ϫ 2͒ 21 n 23 6! 720 25 9! 362,880 27 P4 3024 29 P5 120; 55 3125 8! 7! 11! 31 6720; 2520; 4,989,600 3! 2! 2! 2! 2! 9! 33 и 106 8,000,000; million 35 1260 2! 3! 4! 37 (a) P5 720; (b) 7776; (c) и P4 120; (d) и и P3 24 39 (a) 105, or 100,000; (b) 100,000 41 (a) 10 1,000,000,000; (b) yes 43 Discussion and Writing 44 [2.1] 94 , or 2.25 Ϯ ͙17 45 [2.3] Ϫ3, 46 [2.3] 47 [3.4] Ϫ2, 1, 49 51 11 53 n Ϫ BBEPMN00_0321279115.QXP A-82 1/7/05 3:54 PM Page A-82 Answers Exercise Set 10.6 Exercise Set 10.7 78 78 7 10 11 15 13 128 15 270,725 17 13,037,895 19 n 21 23 23C 8855 25 13C 10 286 8 52 27 29 28; 56 2,598,960 31 (a) 31P2 930; (b) 31 961; (c) 31C 465 33 Discussion and Writing 34 [2.1] Ϫ 172 Ϫ5 Ϯ ͙21 35 [2.3] Ϫ1, 32 36 [2.3] 13 n n 37 [3.4] Ϫ4, Ϫ2, 39 41 ;2 1287 2 43 45 47 Line segments: x ϩ 20x ϩ 150x ϩ 500x ϩ 625 x Ϫ 15x ϩ 90x Ϫ 270x ϩ 405x Ϫ 243 x Ϫ 5x 4y ϩ 10x 3y Ϫ 10x 2y ϩ 5xy Ϫ y 15,625x ϩ 75,000x 5y ϩ 150,000x 4y ϩ 160,000x 3y ϩ 96,000x 2y ϩ 30,720xy ϩ 4096y 128t ϩ 448t ϩ 672t ϩ 560t ϩ 280t Ϫ1 ϩ 84t Ϫ3 ϩ 14t Ϫ5 ϩ t Ϫ7 11 x 10 Ϫ 5x ϩ 10x Ϫ 10x ϩ 5x Ϫ 13 125 ϩ 150͙5 t ϩ 375t ϩ 100͙5 t ϩ 75t ϩ 6͙5 t ϩ t 15 a Ϫ 18a ϩ 144a Ϫ 672a ϩ 2016a Ϫ 4032a Ϫ1 ϩ 5376a Ϫ3 Ϫ 4608a Ϫ5 ϩ 2304a Ϫ7 Ϫ 512a Ϫ9 17 140͙2 19 x Ϫ8 ϩ 4x Ϫ4 ϩ ϩ 4x ϩ x 21 21a 5b 23 Ϫ252x 5y 25 Ϫ745,472a 27 1120x 12y 29 Ϫ1,959,552u 5v 10 31 27, or 128 33 224, or 16,777,216 35 20 37 Ϫ12 ϩ 316i n n 39 Ϫ7 Ϫ 4͙2i 41 ͑Ϫ1͒ka nϪkb k k0 k n n nϪk kϪ1 43 45 Discussion and Writing x h k1 k 46 [1.6] x ϩ 2x Ϫ 47 [1.6] 2x Ϫ 3x ϩ 2x Ϫ 48 [1.6] 4x Ϫ 12x ϩ 10 49 [1.6] 2x Ϫ 51 Ϫ5 Ϯ 2͙2, Ϫ5 Ϯ 2͙2 i 53 3, 9, Ϯ 3i 35 55 Ϫ4320x 6y 9/2 57 Ϫ 1/6 59 2100 61 ͓loga ͑xt͔͒ 23 x 63 (1) Basis step: Since a ϩ b ͑a ϩ b͒1, S is true (2) Induction step: Let S k be the statement of the binomial theorem with n replaced by k Multiply both sides of S k by ͑a ϩ b͒ to obtain ͩͪ ͩͪ ͩͪ ͩͪ nC2 ͩͪ ͩͪ n! n͑n Ϫ 1͒ ͑n Ϫ 2͒! n͑n Ϫ 1͒ 2! ͑n Ϫ 2͒! и и ͑n Ϫ 2͒! Diagonals: The n line segments that form the sides of the n-agon are not diagonals Thus the number of diagonals is n͑n Ϫ 1͒ Ϫn nC2 Ϫ n n2 Ϫ n Ϫ 2n n2 Ϫ 3n 2 n͑n Ϫ 3͒ , n Ն Let D n be the number of diagonals on an n-agon Prove the result above for diagonals using mathematical induction n͑n Ϫ 3͒ Sn : , for n 4, 5, 6, Dn 4и1 S4 : D4 k ͑k Ϫ 3͒ Sk : Dk ͑k ϩ 1͒ ͑k Ϫ 2͒ Skϩ1: D kϩ1 (1) Basis step: S4 is true (a quadrilateral has diagonals) (2) Induction step: Assume Sk Note that when an additional vertex Vkϩ1 is added to the k-gon, we gain k segments, of which are sides of the ͑k ϩ 1͒-gon, and a former side V1Vk becomes a diagonal Thus the additional number of diagonals is k Ϫ ϩ 1, or k Ϫ Then the new total of diagonals is D k ϩ ͑k Ϫ 1͒, or D kϩ1 D k ϩ ͑k Ϫ 1͒ k ͑k Ϫ 3͒ ϩ ͑k Ϫ 1͒ ͑k ϩ 1͒ ͑k Ϫ 2͒ ͩ ͪ ͩ ͪ ͑a ϩ b͒kϩ1 ͫ ak ϩ и и и ϩ ͩͪ ͩ ͪ k a kϪ͑rϪ1͒b rϪ1 rϪ1 ͬ k kϪr r a b ϩ и и и ϩ b k ͑a ϩ b͒ r k k a kϩ1 ϩ и и и ϩ ϩ a ͑kϩ1͒Ϫr b r rϪ1 r ϩ и и и ϩ b kϩ1 k ϩ ͑kϩ1͒Ϫr r a kϩ1 ϩ и и и ϩ a b ϩ и и и ϩ b kϩ1 r This proves S kϩ1, assuming S k Hence S n is true for n 1, 2, 3, ϩ ͫͩ ͪ ͩ ͪͬ ͩ ͪ Exercise Set 10.8 By Sk (a) 0.18, 0.24, 0.23, 0.23, 0.12; (b) Opinions may vary, but it seems that people tend not to pick the first or last numbers 11,700 pieces (a) T, S, R, N, L; (b) E; (c) yes 350 (a) 72 ; (b) 57 ; (c) (d) 31,977 11 108,290 33 13 66,640 15 (a) HHH, HHT, HTH, HTT, THH, THT, TTH, TTT; (b) 83 ; (c) 78 ; (d) 87 ; (e) 38 17 199 19 381 BBEPMN00_0321279115.QXP 1/7/05 3:54 PM Page A-83 Chapter 10 18 21 19 23 199 25 Answers will vary 27 Discussion and Writing 28 [2.1] Zero 29 [4.1] One-to-one 30 [1.2] Function; domain; range; domain; range 31 [2.1] Zero 32 [10.6] Combination 33 [3.7] Inverse variation 34 [3.3] Factor 35 [10.3] Geometric sequence 36 37 (a) 36; (b) Ϸ 1.39 ϫ 10Ϫ5 52C 39 (a) ͑13 и 4C ͒ и ͑12 и 4C ͒ 3744; (b) 0.00144 13 41 (a) и Ϫ Ϫ 36 5108; (b) 0.00197 10 4 4 43 (a) Ϫ Ϫ 36 10,200; 1 1 1 (b) 0.00392 ͩͪ ͩ ͪͩ ͪͩ ͪͩ ͪͩ ͪͩ ͪ Review Exercises: Chapter 10 16 121 529 [10.1] Ϫ 12 , 174 , Ϫ 829 , 257 ; Ϫ 14,642 ; Ϫ 279,842 81 417 [10.1] ͑Ϫ1͒nϩ1͑n ϩ 1͒ [10.1] 32 Ϫ 98 ϩ 27 26 Ϫ 80 1040 1,100,000 [10.1] n Un 10 ͑k [10.1] k1 0.3 2.5 13.5 68.5 343.5 1718.5 8593.5 42968.5 214843.5 1074218.5 Ϫ 1͒ [10.2] 154 10 A-83 1͑3 Ϫ 1͒ is true 2 (2) Induction step: Assume S k Add ͑3k ϩ 1͒ to both sides (1) Basis step: ϩ ϩ ϩ и и и ϩ ͑3k Ϫ 2͒ ϩ ͑3k ϩ 1͒ k͑3k Ϫ 1͒ ϩ ͑3k ϩ 1͒ k͑3k Ϫ 1͒ 2͑3k ϩ 1͒ ϩ 2 3k Ϫ k ϩ 6k ϩ 3k ϩ 5k ϩ ͑k ϩ 1͒ ͑3k ϩ 2͒ 3n Ϫ 24 [10.4] S n : ϩ ϩ 32 ϩ и и и ϩ 3nϪ1 31 Ϫ S 1: 1 3k Ϫ S k : ϩ ϩ 32 ϩ и и и ϩ 3kϪ1 3kϩ1 Ϫ ͑kϩ1͒Ϫ1 S kϩ1: ϩ ϩ ϩ и и и ϩ 31 Ϫ (1) Basis step: is true 2 (2) Induction step: Assume S k Add 3k on both sides ϩ ϩ и и и ϩ 3kϪ1 ϩ 3k 3k Ϫ 3k Ϫ ϩ 3k ϩ 3k и 2 и 3k Ϫ 3kϩ1 Ϫ 2 [10.2] a ϩ 4b [10.2] 531 [10.2] 20,100 10 [10.2] 11 11 [10.2] Ϫ4 12 [10.3] n 6, S n Ϫ126 13 [10.3] a 8, a 12 14 [10.3] Does not exist 15 [10.3] 113 16 [10.3] 38 17 [10.3] 241 99 18 [10.2] 5 , , , 19 [10.3] 167.3 ft 20 [10.3] $45,993.04 21 [10.2] (a) $7.38; (b) $1365.10 22 [10.3] $88,888,888,889 n͑3n Ϫ 1͒ 23 [10.4] S n : ϩ ϩ ϩ и и и ϩ ͑3n Ϫ 2͒ 1͑3 Ϫ 1͒ S 1: 1 k͑3k Ϫ 1͒ S k : ϩ ϩ ϩ и и и ϩ ͑3k Ϫ 2͒ S kϩ1: ϩ ϩ ϩ и и и ϩ ͓3͑k ϩ 1͒ Ϫ 2͔ ϩ ϩ ϩ и и и ϩ ͑3k Ϫ 2͒ ϩ ͑3k ϩ 1͒ ͑k ϩ 1͒ ͑3k ϩ 2͒ ͩ ͩ ͩ ͩ 25 [10.4] Sn : S 2: Sk : S kϩ1: 1Ϫ 1Ϫ 1Ϫ 1Ϫ ͪͩ ͪ ͩ ͪ ͪ ͪͩ ͪ ͩ ͪ ͪͩ ͪ ͩ ͪͩ kϩ1 иии 1Ϫ n n 1Ϫ иии 1Ϫ k k 1Ϫ иии 1Ϫ k 1Ϫ 1Ϫ ͪ kϩ1 BBEPMN00_0321279115.QXP A-84 1/7/05 3:54 PM Page A-84 Answers (1) Basis step: S is true by substitution (2) Induction step: Assume S k Deduce S kϩ1 Starting with the left side of S kϩ1, we have ͩ ͪͩ ͪ ͩ ͪͩ ͩ ͪ ͩ ͪ 1Ϫ 1Ϫ иии 1Ϫ k 1Ϫ 1 и 1Ϫ k kϩ1 ͪ kϩ1 Test: Chapter 10 [10.1] Ϫ43 [10.1] ͩͪ ͩͪ 86 97 23 44 [10.8] 206 Ϸ 0.42, 206 Ϸ 0.47, 206 Ϸ 0.11 45 [10.1] (a) a n 0.1112988224n ϩ 3.300942958; (b) 7.7529 million 46 Discussion and Writing [10.6] A list of candidates for an office is to be narrowed down to candidates In how many ways can this be done? 47 Discussion and Writing [10.3] Someone who has managed several sequences of hiring has a considerable income from the sales of the people in the lower levels However, with a finite population, it will not be long before the salespersons in the lowest level have no one to hire and no one to sell to 48 [10.4] S fails for both (a) and (b) a kϩ1 bkϩ1 a kϩ1bkϩ1 49 [10.3] r1, r2, so r1r2, a constant ak bk a kbk 50 [10.2] (a) No (unless a n is all positive or all negative); (b) yes; (c) yes; (d) no (unless a n is constant); (e) no (unless a n is constant); (f ) no (unless a n is constant) 51 [10.2] Ϫ2, 0, 2, 52 [10.3] 21 , Ϫ 16 , 181 x 10 53 [10.6] log 54 [10.6] 18 55 [10.6] 36 y 56 [10.7] Ϫ9 ͩ ͪ [10.1] ϩ ϩ 10 ϩ 17 34 By Sk kϩ1Ϫ1 и k kϩ1 k и k kϩ1 Simplifying kϩ1 26 [10.5] 6! 720 27 [10.5] и и и 3024 15 6435 29 [10.5] 24 и 23 и 22 12,144 28 [10.6] 9! 3780 31 [10.5] и и 36 30 [10.5] 1! 4! 2! 2! 32 [10.5] (a) P5 720; (b) 65 7776; (c) P4 120; (d) P2 33 [10.7] 28, or 256 34 [10.7] m ϩ 7m 6n ϩ 21m 5n ϩ 35m 4n ϩ 35m 3n ϩ 21m 2n ϩ 7mn ϩ n 35 [10.7] x Ϫ 5͙2 x ϩ 20x Ϫ 20͙2 x ϩ 20x Ϫ 4͙2 36 [10.7] x Ϫ 12x 6y ϩ 54x 4y Ϫ 108x 2y ϩ 81y 37 [10.7] a ϩ 8a ϩ 28a ϩ 56a ϩ 70 ϩ 56a Ϫ2 ϩ 28a Ϫ4 ϩ 8a Ϫ6 ϩ a Ϫ8 38 [10.7] Ϫ6624 ϩ 16,280i 18 39 [10.7] 220a 9x 40 [10.7] Ϫ 128a 7b 11 11 41 [10.8] 121 ; 42 [10.8] 41 43 [10.8] 5525 [10.1] 32 , 43 , 45 , 65 , n Un 10 0.66667 0.75 0.8 0.83333 0.85714 0.875 0.88889 0.9 0.90909 0.91667 4k k1 [10.1] 10 2 ϱ [10.1] k1 k [10.1] 3, 13, 37, 177 [10.2] 44 [10.2] 38 10 [10.2] Ϫ420 11 [10.2] 675 12 [10.3] 512 13 [10.3] 1000 14 [10.3] 510 15 [10.3] 27 16 [10.3] 56 99 17 [10.1] $10,000, $8000, $6400, $5120, $4096, $3276.80 18 [10.2] $12.50 19 [10.3] $74,399.77 20 [10.4] n͑3n ϩ 1͒ Sn : ϩ ϩ ϩ и и и ϩ ͑3n Ϫ 1͒ 1͑3 и ϩ 1͒ S 1: 2 k͑3k ϩ 1͒ Sk : ϩ ϩ ϩ и и и ϩ ͑3k Ϫ 1͒ Skϩ1: ϩ ϩ ϩ и и и ϩ ͑3k Ϫ 1͒ ϩ ͓3͑k ϩ 1͒ Ϫ 1͔ ͑k ϩ 1͒ ͓3͑k ϩ 1͒ ϩ 1͔ 1͑3 и ϩ 1͒ и (1) Basis step: 2, so S1 is true 2 (2) Induction step: ϩ ϩ ϩ и и и ϩ ͑3k Ϫ 1͒ ϩ ͓3͑k ϩ 1͒ Ϫ 1͔ k͑3k ϩ 1͒ ϩ ͓3k ϩ Ϫ 1͔ k 3k ϩ ϩ 3k ϩ 2 3k 7k ϩ ϩ2 2 3k ϩ 7k ϩ ͑k ϩ 1͒ ͑3k ϩ 4͒ ͑k ϩ 1͒ ͓3͑k ϩ 1͒ ϩ 1͔ BBEPMN00_0321279115.QXP 1/7/05 3:54 PM Page A-85 Chapter 10 – Appendix 21 [10.5] 3,603,600 22 [10.6] 352,716 n͑n Ϫ 1͒ ͑n Ϫ 2͒ ͑n Ϫ 3͒ 23 [10.6] 24 [10.5] P4 360 24 25 [10.5] (a) 64 1296; (b) P3 60 26 [10.6] 28C 20,475 27 [10.6] 12C и 8C 34,650 28 [10.7] x ϩ 5x ϩ 10x ϩ 10x ϩ 5x ϩ 29 [10.7] 35x 3y 30 [10.7] 29 512 31 [10.8] 48 32 [10.8] 1001 33 [10.5] 15 Appendix Exercise Set Acute Straight Obtuse 79Њ 23Њ 11 32Њ 13 61Њ 15 177Њ 17 41Њ 19 95Њ 21 78Њ 23 ЄR i ЄA, ЄS i ЄB, ЄT i ЄC ; RS i AB, RT i AC , ST i BC A-85 25 ЄC i ЄW , ЄB i ЄJ , ЄS i ЄZ ; CB i WJ , CS i WZ , BS i JZ AB AC BC RS RT ST ME MS ES ЄM i ЄC , ЄE i ЄL, ЄS i ЄF ; CL CF LF PS SQ PQ TA TW AW 33 ND DM NM GF GC FC 39 36 ft 41 17 QR 10, PR 37 100 ft ͙32 Ϸ 5.657 45 12 47 49 26 12 53 55 ͙2 Ϸ 1.414 57 59 63 240 ft ͙211,200,000 ft Ϸ 14.533 ft 67 ͙208 ft Ϸ 14.422 ft ͙18 cm Ϸ 4.243 cm 27 ЄA i ЄR, ЄB i ЄS, ЄC i ЄT ; 29 31 35 43 51 61 65