MAT101 General Mathematics 1 Study Guide PDF
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Olabisi Onabanjo University
2021
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This is a study guide for MAT101 General Mathematics 1, a 3-unit course at Olabisi Onabanjo University. The course covers fundamental mathematical topics like set theory, Venn diagrams, number systems, mathematical induction, sequences, series, quadratic equations, and more. The course is designed for accounting undergraduates.
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MAT101: General Mathematics 1 OLABISI ONABANJO UNIVERSITY OPEN AND DISTANCE LEARNING CENTRE AGO IWOYE MAT101: GENERAL MATHEMATICS I Page 1 of 164 MAT101: General Mathematics 1 MAT102 STUDY GU...
MAT101: General Mathematics 1 OLABISI ONABANJO UNIVERSITY OPEN AND DISTANCE LEARNING CENTRE AGO IWOYE MAT101: GENERAL MATHEMATICS I Page 1 of 164 MAT101: General Mathematics 1 MAT102 STUDY GUIDE Introduction MAT101 titled General Mathematics I is a 3 unit course for students studying towards acquiring a Bachelor of Science in Accounting. The course is divided into 13 study sessions. The course will introduce you to the basic mathematics concept in solving practical problems. The course study guide therefore gives you an overview of what MAT101 is all about, the textbooks and other materials to be referenced, what you are expected to know in each unit and how to work through the course materials. Define a set and identify various notations of sets. Explain the two ways of describing sets. Identify and define various types of set operations with their applications. State the difference between union and intersection of a set. Define the difference between two sets. Define a singleton set. Recommended Study Time This course is a 3 unit course divided into 8 study sessions. You are enjoined to spend at least 3 hours in studying the content of each study unit What you are about to learn in this course The overall aim of this course, MAT101 is to introduce you to Set Theory, Venn Diagram, Number System, Mathematical Induction, Real Sequences and Series, Theory of Quadratic Equation, The Binomial Theorem, Complex Number, Circle Geometry, Parametric Equation of a Circle and Trigonometry Course Aims This course aims to introduce students to the basic Mathematical concept of Set functions, matrices and different methods of solving simultaneous equations. It is expected that the knowledge will help the reader to effectively use Mathematics principles to solve even life problems. Course Objectives It is important to note that each unit has specific objectives. You should study them carefully before proceeding to subsequent units. Therefore, it may be useful to refer to these objectives in the course of your study of the unit to assess your progress. You should always look at the unit objectives after completing a unit. In this way, you can be sure that you have done what is required of you by the end of the unit. Page 2 of 164 MAT101: General Mathematics 1 However, the followings are overall objectives of this course. On completing this course, you should be able to: 1 Define a set and identify various notations of sets 2 Explain the two ways of describing sets. 3 Identify and define various types of set operations with their applications. 4 Define the difference between two sets. 5 Identify and define various types of set operations with their applications. 6 Illustrate set operations using Venn diagrams. 7 Make use of Venn diagrams. 8 Relate the solutions in set to real life problems. 9 Identify and define various types of set operations with their applications. 10 Illustrate set operations using Venn diagrams. 11 Make use of Venn diagrams. 12 Relate the solutions in set to real life problems. 13 Define Mathematical induction 14 Use Mathematical induction to prove statement for natural numbers 15 Understand what a sequence means 16 Calculate the arithmetic progression and arithmetic mean of a sequence 17 Solve the geometric progression and geometric mean of a sequence 18 Find the sum of arithmetic progression and geometric progression of a sequence. 19 Solve quadratic equation by factorizing, completing the square and by using the quadratic formula. 20 Find the sum and products of the roots without solving the quadratic equation 21 Expand using Pascal’s triangle 22 Apply binomial theorem 23 Understand and classify a complex number 24 Find the conjugate of a complex number 25 Perform different operations on a complex number 26 Represent a complex number on the Argand diagram and add and subtract a complex number graphically. 27 Find the modulus and argument of a complex number 28 Write a complex Number in polar form and carry out operations in complex form 29 Write a complex number in exponential form. Page 3 of 164 MAT101: General Mathematics 1 30 Understand and apply De Moivre’s Theorem and the Nth root of unity. 31 State the general form of equation of a circle 32 Solve some problems on the equation of a circle. 33 Proffer solutions to equation of the tangent and normal to a circle. 34 State the parametric equation from a given circle equation 35 Solve some problems on the parametric equation of a circle given the general equation of a circle. 36 Determine the centre and radius of the circle given the parametric equation of a circle. 37 State some formulas on circular functions. 38 Use trigonometric identities to manipulate and prove certain trigonometric problems. 39 Use the concept of trigonometry in solving practical problems as it applies to elevation and depression and triangles 40 State some trigonometric identities 41 Prove certain trigonometric identities. 42 Use the concept of trigonometry in solving practical problems as it applies to elevation and depression and triangles Working through this course In order to have a thorough understanding of the course units, you will need to read and understand the contents, practice the steps by designing and implementing a mini computer application system for your department and be committed to learning and implementing your knowledge. This course is designed to cover approximately fifteen weeks and it will require your devoted attention. You should do the exercises in the Tutor-Marked Assignments and submit to your tutors via the LMS. Course Materials The major components of the course are; 1. Course Guide 2. Printed Lecture materials 3. Text Books 4. Interactive DVD 5. Electronic Lecture materials via LMS Page 4 of 164 MAT101: General Mathematics 1 6. Tutor Marked Assignments Printed Lecture Materials The printed lecture material consists of 8 study sessions broken down into sub-sessions; Study Session One: Set Theory 1.1 Define a set and identify various notations of sets 1.2 Explain the two ways of describing sets. 1.3 Identify and define various types of set operations with their applications. 1.4 Define the difference between two sets. Study Session Two: The Venn Diagram 2.1. Identify and define various types of set operations with their applications. 2.2. Illustrate set operations using Venn diagrams. 2.3. Make use of Venn diagrams. 2.4. Relate the solutions in set to real life problems. Study Session Three: Number System 3.1 Identify and define various types of set operations with their applications. 3.2 Illustrate set operations using Venn diagrams. 3.3 Make use of Venn diagrams. 3.4 Relate the solutions in set to real life problems. Study Session Four: Mathematical Induction 4.1 Define mathematical induction 4.2 Use mathematical induction to prove statement for natural numbers Study Session Five: Real Sequence and Series 5.1 Understand what a sequence means 5.2 Calculate the arithmetic progression and arithmetic mean of a sequence 5.3 Solve the geometric progression and geometric mean of a sequence Find the sum of arithmetic progression and geometric progression of a sequence Study Session Six: Theory of Quadratic Equation 6.1 Solve quadratic equation by factorizing, completing the square and by using the quadratic formula. Page 5 of 164 MAT101: General Mathematics 1 6.2 Find the sum and products of the roots without solving the quadratic equation. Study Session seven: The Binomial Theorem 7.1 Expand using Pascal’s triangle 7.2 Apply binomial theorem Study Session 8: Complex Number 1 8.1 Understand and classify a complex number 8.2 find the conjugate of a complex number 8.3 perform different operations on a complex number 8.4 represent a complex number on the Argand diagram and add and subtract a complex number graphically. Study Session 9: Complex Number 2 9.1 Find the modulus and argument of a complex number 9.2 Write a complex Number in polar form and carry out operations in complex form 9.3 Write a complex number in exponential form. 9.4 understand and apply De Moivre’s Theorem and the Nth root of unity. Study Session 10: Circle Geometry 10.1 State the general form of equation of a circle 10.2 Solve some problems on the equation of a circle. 10.3 Proffer solutions to equation of the tangent and normal to a circle. Study Session 11: Parametric Equations of a Circle 11.1 State the parametric equation from a given circle equation 11.2 Solve some problems on the parametric equation of a circle given the general equation of a circle. 11.3 Determine the centre and radius of the circle given the parametric equation of a circle. Study Session 12: Trigonometry 12.1State some formulas on circular functions. 12.2 Use trigonometric identities to manipulate and prove certain trigonometric problems. Page 6 of 164 MAT101: General Mathematics 1 12.3 Use the concept of trigonometry in solving practical problems as it applies to elevation and depression and triangles. Study Session 13: Trigonometric Identities 13.1 State some trigonometric identities 13.2 Prove certain trigonometric identities. 13.3 Use the concept of trigonometry in solving practical problems as it applies to elevation and depression and triangles Recommended Texts The following texts and Internet resource links will be of enormous benefit to you in learning this course: 1. Nwagbogwu, D. C. and Akinfenwa, O. A. (2008). Fundamentals of Mathematics, S-S Stephen’s Nig. Ltd., Lagos, Nigeria. 2. www.oneaccess.com.ng/results_by_category.php?CategoryID. 3. Matthews, K. R. (1998). Elementary Linear Algebra, Department of Mathematics, University of Queensland. 4. www.math.fsu.edu/~dli/matthews.pdf 5. Anthony Barcellos, (1992). Calculus and Analytic Geometry, Fifth edition, Volume 1, American River College, Sacramento, California. 6. www.amazon.com ›... › Science & Mathematics › Mathematics › Calculus 7. James Stewart, (1999). Calculus (Early Transcendentals), Fourth Edition, McMaster University, U.S.A. 8. www.mybookezz.com/steward-calculus-fourth-edition/ 9. Usman M. A., Odetunde O.S., Ogunwobi Z.O., Hammed F.A. (2016). Mathematics for University Students, Volume 1 Assessment There are two aspects to the assessment of this course. First, there are tutor marked assignments and second, the written examinations. Therefore, you are expected to take note of the facts, information and problem solving gathered during the course. The tutor marked assignments must be submitted to your tutor for formal assessment in accordance to the deadline given. The work submitted will count for 30% of your total course mark. Page 7 of 164 MAT101: General Mathematics 1 At the end of the course, you will need to sit for a final written examination. This examination will account for 70% of your total score. You will be required to submit some assignments by uploading them to MAT101 page on the LMS. Tutor-Marked Assignment (TMA) There are TMAs in this course. You need to submit all the TMAs. The best 10 will therefore be counted. When you have completed each assignment, send them to your tutor as soon as possible and make certain that it gets to your tutor on or before the stipulated deadline. If for any reason you cannot complete your assignment on time, contact your tutor before the assignment is due to discuss the possibility of extension. Extension will not be granted after the deadline, unless on extraordinary cases. Final Examination and Grading The final examination for MAT101 will last for a period not more than 2hours and has a value of 70% of the total course grade. The examination will consist of questions which reflect the Self-Assessment Questions (SAQs), In-text Questions (ITQs), some applied questions and tutor marked assignments that you have previously encountered. Furthermore, all areas of the course will be examined. It would be better to use the time between finishing the last unit and sitting for the examination to revise the entire course. You might find it useful to review your TMAs and comment on them before the examination. The final examination covers information from all parts of the course. Most examinations will be conducted via Computer Based Testing (CBT) Tutors and Tutorials There are few hours of face-to-face tutorial provided in support of this course. You will be notified of the dates, time and location together with the name and phone number of your tutor as soon as you are allocated a tutorial group. Your tutor will mark and comment on your assignments, keep a close watch on your progress and on any difficulties you might encounter and provide assistance to you during the course. You must submit your tutor marked assignment to your tutor well before the due date. At least two working days are required for this purpose. They will be marked by your tutor and returned as soon as possible via the same means of submission. Page 8 of 164 MAT101: General Mathematics 1 Do not hesitate to contact your tutor by telephone, e-mail or discussion board if you need help. The following might be circumstances in which you would find help necessary: contact your tutor if: You do not understand any part of the study units or the assigned readings. You have difficulty with the self-test or exercise. You have questions or problems with an assignment, with your tutor’s comments on an assignment or with the grading of an assignment. You should endeavour to attend the tutorials. This is the only opportunity to have face-to-face contact with your tutor and ask questions which are answered instantly. You can raise any problem encountered in the course of your study. To gain the maximum benefit from the course tutorials, have some questions handy before attending them. You will learn a lot from participating actively in discussions. Good luck! Page 9 of 164 MAT101: General Mathematics 1 Study Session One: Set Theory 1.1 Introduction The concept of set theory is one of the most fundamental concepts in Mathematics. It cuts across every aspect of life. In our activities, we often arrange or group certain things of the same kind together. This act of putting like things together is called set theory. For example, we speak of a football team, a pack of cards, a group of students and so on. These examples are called sets. We see in the above examples a clear connection of objects of the set, but this does not necessarily have to be so. For example, we may have a set containing a bag, a cup and a shirt. The important thing here is that, given any object, we must be able to identify whether or not the object belongs to a given set. Learning Outcomes for Study Session One When you have studied this session, you should be able to: 1.1 Define a set and identify various notations of sets 1.2 Explain the two ways of describing sets. 1.3 Identify and define various types of set operations with their applications. 1.4 Define the difference between two sets. Page 10 of 164 MAT101: General Mathematics 1 Table of Contents MAT101 STUDY GUIDE 2 Study Session One: Set Theory 10 1.1 Introduction 10 Learning Outcomes for Study Session One 10 1.1 Definition of Set 16 1.2 Description of Sets 17 1.2.1 Tabular form 17 1.2.2 Set Builder Form 18 1.2.3 Standard Sets 19 1.2.4 The Universal Set 19 1.2.5 Subset 19 1.2.6 Proper Set 20 1.2.8 Equality of Sets 21 1.3 Types of Sets 22 1.3.1 Singleton Set 22 1.3.2 Finite and Infinite Sets 23 1.4 Set Operations 24 1.4.1 Union of a Set 24 1.4.2 Intersection of a Set 25 1.4.3 Complement of a Set 26 1.4.4 Relative Complement of a Set 26 1.4.5 Symmetric Difference 27 References 28 Summary of Study Session 1 29 Glossary of Terms 30 Self-Assessment Questions (SAQs) for Study Session 1 31 SAQ 1.1 (Objective Questions) 31 SAQ 1.2 (Theory Questions) 32 Study Session 2: The Venn Diagram 33 Introduction 33 Learning Outcomes for Study Session Two 33 2.1 Set Operations 34 2.1.1 Power Set 35 2.1.2 Algebra of Set 36 Page 11 of 164 MAT101: General Mathematics 1 2.1.3 Cardinality of Sets 39 2.2 Application of Set theory 40 Summary of Study Session 2 46 Self-Assessment Questions (SAQs) for Study Session 2 47 SAQ 2.1 47 SAQ 2.2 47 Study Session 3: Number System 49 Introduction 49 Learning Outcomes for Study 3 49 3.1 The Real Number System 50 In-text Questions (ITQs) 3.1 50 In-text Answer (ITAs) 3.1 50 3.2 Properties of Natural Numbers N 51 3.2.1 Addition 51 3.2.2 Multiplication (.) 51 3.2.3 Addition is associative 51 3.2.4 Addition is Commutative 52 3.2.5 Distributivity of Multiplication over Addition 52 Multiplication is Associative 53 3.2.7 Multiplication is commutative. 53 3.2.8 Trichotomy Law 54 3.2.9 Integer 54 3.2.10 Rational Numbers 54 3.2.11 Irrational Numbers 54 3.2.12 Operations on Real Numbers 54 3.2.13 Operation Involving Zero 54 3.2.14 Multiplication by ‘-1’ 55 3.2.15 Operations involving -1 as Index 55 Self-Assessment Questions(SAQs) 58 SAQ 3.1 58 SAQ 3.2 58 References 59 Study Session 4: Mathematical Induction 60 Introduction 60 Learning Outcomes for Study Session 4 60 4.1 Definition 61 Page 12 of 164 MAT101: General Mathematics 1 4.2 Proving Statements with Mathematical Induction 61 4.2.1 Strong Induction 63 Summary 65 Self-Assessment Questions (SAQs) 66 Study Session 5: Real Sequences and Series 67 Introduction 67 Learning Outcomes for Study Session 5 67 5.1 Sequence as a Function 68 5.2 Arithmetic Sequence 69 5.2.1 Arithmetic Progression (A.P.) 69 5.2.2 Arithmetic Mean 71 5.3 Geometric Sequence 72 5.3.1 Geometric Progression (G.P.) 72 5.3.2 Geometric Mean 75 5.4 Series 76 5.4.1 Sum of an Arithmetic Progression 77 5.4.2 Sum of a Geometric Progression 78 5.4.3 Recurrence 79 Summary of Study Session 5 81 Self-Assessment Questions (SAQs) 82 SAQ 5.1 82 SAQ 5.2 82 SAQ 5.3 82 SAQ 5.4 82 Study Session 6: Theory of Quadratic Equation 84 Introduction 84 Learning Outcomes for Study Session 6 84 6.1 Methods of Solving Quadratic Equation 85 6.1.1 Solving Quadratic Equation by Factorizing 85 6.1.2 Square Roots Method 85 6.1.3 Completing the Square Method 86 6.1.4 The Quadratic Formula 87 6.2 Operations on the Roots 87 6.2.1 Sum and Products of the Roots 88 6.2.2 Symmetric Functions of the Roots 88 Summary of study Session 6 91 Page 13 of 164 MAT101: General Mathematics 1 Self-Assessment Questions (SAQs) for Study Session 6 92 Study Session 7: The Binomial Theorem 93 Introduction 93 Learning Outcomes for Study Session 7 93 7.1 Pascal’s Triangle 94 7.2 Binomial Theorem 95 Summary of study Session 7 98 Self-Assessment Questions (SAQs) for Study Session 7 99 Study Session 8: Complex Numbers 1 100 Introduction 100 Learning Outcomes for Study Session 8 100 8.1 Definition 101 8.1.1 Classification of Complex Numbers 102 8.2 Conjugate of a Complex Number 103 8.3 Operations in Complex Numbers 103 8.3.1 Addition and Subtraction 103 8.3.2 Multiplication and Division of Complex Numbers 104 8.3.3 Powers of i 105 8.3.4 Square Roots 105 8.4 The Argand Diagram 107 8.4.1 Graphical Addition of Complex Numbers 107 Summary of study Session 8 109 Self-Assessment Questions (SAQs) for Study Session 8 110 Study Session 9: Complex Number 2 111 Introduction 111 Learning Outcomes for Study Session 9 111 9.1 Modulus and Argument of Complex Number 112 9.1.1 Modulus (or Absolute Value) of Complex Number 112 9.1.2 Argument of a Complex Number 112 9.2 Polar Form of a Complex Number 115 9.2.1 Multiplication of Complex Numbers in the Polar Form 117 9.2.2 Division of Numbers in Polar Form 118 9.3 Exponential Form of a Complex Number 118 9.4 Moivre’s Theorem 119 9.4.1 Nth Root of Unity 119 Summary of study Session 9 120 Page 14 of 164 MAT101: General Mathematics 1 Self-Assessment Questions (SAQs) for Study Session 9 121 Study Session 10: Circle Geometry 122 Introduction 122 Learning Outcomes for Study Session 10 122 10.1 Equation of a Circle 123 10.2 The General Form of Equation of a Circle 126 10.2.1 Characteristics of an Equation of a Circle 126 10.3 Tangent to a Circle 130 10.3.1 Equation of the Tangent to the General Equation of a Circle 131 10.3.2 Normal to a Circle 132 10.3.3 Equation of the Normal to a Circle 132 10.3.4 Equation of the Normal to the General Equation of a Circle 133 Summary of Study Session 10 135 Self Assessment Questions (SAQs) 135 Study Session 11: Parametric Equations of a Circle 137 Introduction 137 Learning Outcomes for Study Session 11 137 11.1 Parametric Equations of a Circle taken the Centre and Radius 138 Self Assessment Questions (SAQs) 142 Study Session 12: Trigonometry 143 Introduction 143 Learning Outcomes for Study Session 12 144 12.1 Sine, Cosine and Tagent of Angles 145 12.2 Rule of Signs 148 12.3 Elementary Angles (Special Angles) 149 12.4 Degrees and Radians 152 Self-Assessment Questions (SAQs) 154 Study Session 13: Trigonometric Identities 156 Introduction 156 Learning Outcomes for Study Session 13 156 13.1 Some trigonometric identities 157 13.2 Double Angles 161 Summary of Study Session 13 163 Self-Assessment Questions (SAQs) 164 Page 15 of 164 MAT101: General Mathematics 1 1.1 Definition of Set A set is a collection or a class of well defined objects. Such objects could be living or non- living. The objects are usually called elements or members of the set. Case Study 1.1 (i). The set of all letters of the alphabet, i.e., {a, b, c, …z} (ii). The set of all integers, i.e., {…, -2, -1, 0, 1, 2, …} (iii). The set of chairmen in all the Local Government Areas of Lagos State (iv). The set of books in the Faculty of Business Administration Library (v). The set of elected presidents and military heads of state of the Federal Republic of Nigeria. Sets are usually denoted by capital letters, and the elements or members are denoted by small letters. Case Study 1.2 If d is an element of the set S then we write d S, which reads “d belongs to S” or “d is a member of S.” If d is not an element of the set S, then we simply write d S, showing that d is not a member of the set S. Now, write the following in symbolic form: a is an element of set A b is not a member of set B Solution You see, it is very easy and short using symbols to show any member of a set. So, if you have written the above in symbolic form, your answer should look like the following: (i) a A (ii) b B In-Text Question 1.1 Given a set E = {set of prime numbers} such that e E. Which of the following is NOT true about set E? Page 16 of 164 MAT101: General Mathematics 1 A. 2 E B. 4 E C. 11 E D. 3 E In-Text Answer 1.1 B 1.2 Description of Sets There are two ways of describing a set. These are (i) tabular form and (ii) set builder form. 1.2.1 Tabular form This involves the listing of elements which make up the set, with each element separated by a comma before writing another element, then enclosing the elements within braces. For instance, if C is a set consisting of the first five positive integers, then we can write: C = {1, 2, 3, 4, 5} or C = {1, 2, …, 5}. Case Study 1.3 Q = {a, b, c, d} Here, Q is the set with elements a, b, c, d. Now, you can write the following in tabular form using any letter of your choice to denote the set. The set of positive odd integers The set of elected presidents or military heads of state of the Federal Republic of Nigeria till 2013. Solution All you need do is to list the elements of the set and enclose them in curly brackets or braces which we believe you must have done successfully; but you may check your answers with the following: A = {1, 3, 5, 7, …}. The dots show that set A is an infinite set and the elements still continue indefinitely. P = {Dr.NnamdiAzikiwe, Major AgunyiIronsi, General Yakubu Gowon, General Muritala Mohammed, General Olusegun Obasanjo, Major General Mohammed Buhari, General Ibrahim Babangida, Chief Earnest Shonekan, General Sanni Abacha, General AbdulsalamiAbubakar, Chief Olusegun Obasanjo, Alhaji Umar Musa Yaradua, Dr.GoodluckEbele Jonathan}. Page 17 of 164 MAT101: General Mathematics 1 1.2.2 Set Builder Form This gives the precise property or properties characterising each element of the set. It has the form: S = {x | x possesses the property, Y}, where x in this case is an arbitrary element of the set S and the symbol “|” denotes “such that.” Case Study 1.4 Write the following in set builder form using any symbol of your choice. The set of months of the year The set of students in DLI Solution M = {x | x is a month of the year} B = {x | x is a student in DLI} From the solution above you can see that the arbitrary element x gives the property of the set, i.e., it tells you what set of people or things are being described. 1. a) Let A = y | y is a prime number. List the elements of set A. b) Let B = Types of triangle. List the elements of set B. c) Let C = Multiples of 5 up to 25. List the elements of set C. d) Let D = y | y is a factor of 24, y is greater than 10. List the elements of set D. 2. a) Let E = 2, 4, 6, 8,... Page 18 of 164 MAT101: General Mathematics 1. Use set builder notation to describe set E. b) Let F = vertebrates, invertebrates, flowering plants, non-flowering plants. Use set builder notation to describe set F. c) Let G = Mercury, Venus, Earth, Mars, Jupiter, Uranus, Saturn, Neptune, Pluto. Use set builder notation to describe set G. d) Let H = 1, 8, 27, 64, 125,.... Use set builder notation to describe set H. 1.2.3 Standard Sets The list of standard notations for sets is as follows: = the set of integers {0, 1, 2, 3, …} a ℚ= the set of rational numbers ,b 0 b ℝ= the set of real numbers { + ℚ} + = the set of positive integers. This set is also called the set of natural numbers and is denoted by. ℝn= Euclidean n-space, e.g., X = (X1, X2, …, Xn) Ȼ= the set of complex numbers 1.2.4 The Universal Set The universal set is the set that contains all elements under consideration in a particular problem. It is denoted by ξ or. Case Study 1.5 If A = {1, 3, 5}, B = {x, y, z}, then the universal set is U = {1, 3, 5, x, y, z}, which implies all the elements in both sets A and B. 1.2.5 Subset Consider the students who sit in the front row of the class and the entire students in the class. The first category of students form part of the second but not vice-versa. Page 19 of 164 MAT101: General Mathematics 1 Definition 1.1 Let A and B be two sets, such that every element of A is an element of B. Then, we say A is a subset of B and we write A B or B A which reads “A is contained in B” or “B contains A.” In some texts, the subset is denoted by or . If A is not a subset of B, we write A B (which reads A is not contained in B). Case Study 1.6 Let X = {a, b, c, 1, 2}, Y = {a, b, c, d, 1, 2, 3}, Z = {c, d, e, 1, 2, 3}. Which of these sets are subsets of the other? Solution Since subset is a part of another set, then, from the above, X Y but X Z i.e., X is a subset of Y but X is not a subset of Z. 1.2.6 Proper Set Definition 1.2 A set A is said to be a proper subset of a set B if (i) A is a subset of B (ii) A is not equal to B Case Study 1.7 Which of these sets are proper subsets of the other? X = {2, 5, 7}, Y = {2, 4, 5, 7} and Z = {2, 3, 4, 5, 6, 7} Solution Here, we are expected to look for the set whose every member is a member of another set. Thus, we see that X is a proper subset of Y and Z, so we write X Y and X Z. Let X = {a, b, c}, Y = {a, b, c, d, 1, 3, 7} and Z = {a, b, 1, 3}. (i) Write the universal set for the sets. (ii) Which of these sets are subsets of the others? Page 20 of 164 MAT101: General Mathematics 1 (iii) Which of these are proper subsets of the others? (i) U = {a, b, c, d, 1, 3, 7} (ii) X Y and Z Y i.e., X is a subset of Y and Z is a subset of Y (iii) X Y and Z Y i.e., X is a proper subset of Y and Z is a proper subset of Y. 1.2.8 Equality of Sets Two sets A and B are said to be equal if and only if every element of A is a member of B and every element of B is a member of A. Then we write A = B Logically speaking, A = B means (a A = a B) or the Bi-conditional statement (a A) (a B) is true for all a. Case Study 1.8 Let A and B be two non-empty sets. If A = {a, b} and B = {a, b}, we say A B and B A, then A = B So, you can say A is equal to B if A is contained in B and B is contained in A. Now, if X = {1, 2, 3}, Y = {1, 2, 3, 4} and Z = {1, 2, 3}. Is (i). X equal to Y? (Why?) (ii). X equal to Z? (Why?) Solution Your answer is as good as mine, but see if we answered the same way. (i) No, X is not equal to Y, i.e., X Y, because every element of X is a member of Y, but every element of Y is not a member of X, e.g., 4 is not a member of X. (ii) Yes, X is equal to Z, because every element of X is a member of Z and every element of Z is a member of X, i.e., X Z, Z X. Therefore, X = Z. Page 21 of 164 MAT101: General Mathematics 1 1.3 Types of Sets There are different types of sets. These include empty or null set, singleton, finite and infinite set, and power set. Definition 1.2 Null Set: A null or empty set is denoted by or { }; it is a set which has no element. Case Study 1.9 Let P be the set of all real numbers whose squares is –1, then P = since there is no real number whose square is –1. Case Study 1.10 Let B = {x | x is both even and odd} Then, B = since there is no integer that is both even and odd. 1.3.1 Singleton Set A set containing only one element is called a singleton set. Case Study 1.11 Y = {x | x is the current president of Nigeria} In this example, the only element of the set is Dr.Goodluck Ebele Jonathan who is the current President of Nigeria, so we write: Y = {Dr.GoodluckEbele Jonathan} Case Study 1.12 If the set A is given as A = {a}, can you say A is a singleton set? Why? Solution Yes. A is a singleton set because ‘a’ is the only element of the set. Page 22 of 164 MAT101: General Mathematics 1 1.3.2 Finite and Infinite Sets A set is said to be finite if it consists a countable or definite number of elements. If the number of elements in the set is uncountable or indefinite, then we say the set is an infinite set. Case Study 1.13 Let D be the set of days of the week. Then, D = {Monday, Tuesday, …, Sunday} is a finite set. So you observe that we have 7 days in a week. Since the set D has 7 elements, we say D is a finite set. Case Study 1.14 Let N be the set of natural numbers. Then, N = {1, 2, 3, …} is an infinite set because it contains an uncountable number of elements. The dots … show that the elements of the set continue indefinitely. What can you say of the following sets: M = {x | x is a natural number less than 20} P = {3, 9, 15, 21, …} B = {John, James, Paul, Peter} X = {x | x is from Lagos State} Z = {x | x is a grain of sand} Q = {q} Well done! Your opinion for each of the questions is quite encouraging. See if they correspond to the ones below. M = {1, 2, 3, …, 19} is finite because the number of elements of M is known. P = {3, 9, 15, 21, …} is infinite. P is the set of numbers divisible by 3 and the elements of the set continue indefinitely. B = {John, James, Paul, Peter}is finite because the set B has 4 elements. We can also say B is the set of Christians that are males. Page 23 of 164 MAT101: General Mathematics 1 1.4 Set Operations In elementary arithmetic, addition, multiplication, division and subtraction are operations which are also valid on the set of numbers. These then enable us to combine two elements of sets to form a new set. To this end, we shall define the operations of sets, which include union, intersection, complement, relative complement, and symmetric difference. 1.4.1 Union of a Set The union of a set denoted by is like addition in numbers. It combines all the elements of the sets in question without repeating any of the elements. For instance, let A and B be two sets. The union of A and B, denoted by A B, is the set of elements which belong to either A or B or both. That is, the combination of elements belonging to A and B without repetition. A B can also be defined as A B = {x | x A or x B}. Now, x A means x A B, by definition A A B. And x B means x A B, by definition B A B. Case Study 1.15 If A = {2, 5, 8} and B = {3, 4, 5, 6, 7}, find A B. Solution Since A B is the combination of elements of A and B, then, A B = {2, 3, 4, 5, 6, 7, 8} Observe that elements of A are contained in A B and we write A A B. Also, elements of B are contained in A B and we write B A B. Hence, A A B and B A B. Find the union of the following pairs of sets. A = {a, b, c}and B = {a, e, o, u} X = {3, 6, 9, 12} and Y = {2, 4, 6, 8, 10} Well done! Very easy now, isn’t it? You may check if you made a mistake. (i) A B = {a, b, c, e, o, u} (ii) X Y = {2, 3, 4, 6, 8, 9, 10, 12} Page 24 of 164 MAT101: General Mathematics 1 1.4.2 Intersection of a Set Here, we talk of elements which are common to two or more sets. The intersection of sets, denoted by , is therefore all the elements common to any two or more sets. For instance, let A and B be any two sets. Their intersection is denoted by A B and is the set of all elements which are common to sets A and B. A B can also be defined as: A B = {x | x A and x B} Note that OR is used for union while AND is used for intersection. Case Study 1.16 Given A = {a, e, i, o, u} and B = {a, b, c, d, e, f}, find A B. Solution Since intersection comprises all the elements common to both A and B, then we have that A B = {a, e} Case Study 1.17 1 1 1 1 1 2 4 , , , , ,1, If A = 2 3 4 5 and B = 3 3 3 , what is A B? Solution 1 A B = 3. Observe that only elements present in A and B show their intersection. So, this property is also for more than two sets. We only take the common elements in all the sets to form the set of their intersection. If A and B have no elements in common, then we write A B = . If this happens, we say that the sets A and B are disjoint. Therefore, we say two or more sets are disjoint if their intersection is empty. Page 25 of 164 MAT101: General Mathematics 1 1.4.3 Complement of a Set In this case, we compare any set with respect to the universal set. That is, we identify those elements which are in the universal set but are not in the set of interest. Therefore, the complement of a set A denoted by AC or A' or C(A) are those elements in the universal set but not in set A, and we write, AC = 𝒰 – A = {x | x 𝒰 but x A} Case Study 1.18 Let 𝒰 = {1, 2, …, 10} and A = {1, 3, 5, 7, 9}. Find the complement of A. Solution Since AC is the set of all the elements in the universal set that are not in A, we have, AC = 𝒰 – A = {2, 4, 6, 8, 10}. All we did was to remove all the elements of set A from the element of the universal set 𝒰 to give the complement of set A. 1.4.4 Relative Complement of a Set Here, like the arithmetic subtraction, we subtract one set from another to get their difference. Hence, the relative complement of a set A with respect to another set B, denoted by B – A is the set of all elements in set B that are not in set A. We write B – A = {x | x B and x A} That is, the element x is a member of B but not a member of A. Page 26 of 164 MAT101: General Mathematics 1 Case Study 1.19 If A = {1, 4, 7, 8} and B = {3, 4, 6, 9}, find (i) A – B and (ii) B–A Solution All we need do is to write down all the elements that are in one set but not in the other. A – B = {1, 7, 8} B – A = {3, 6, 9} Note that A – B = A BC and so it is for any relative complement. 1.4.5 Symmetric Difference Let A and B be two sets. Then, the union of the relative complement of set A with respect to set B and the relative complement of set B with respect to set A, is called symmetric difference of the two sets and it is denoted by A B. In other words, the symmetric difference of two sets A and B is the relative complement of the set A B with respect to A B that is, A B = (A – B) (B – A) = (A B) – (A B) Example 1.20 Let P = {0, 1, 2, 5, 7, 8, 15, 20} and Q = {3, 4, 6, 7, 9, 11, 13, 17}. Find P Q. Solution Now, P – Q = {0, 1, 2, 5, 8, 15, 20} and Q – P = {3, 4, 6, 9, 11, 13, 17} Therefore, P Q = (P – Q) (Q – P) = {0, 1, 2, 3, 4, 5, 6, 8, 9, 11, 13, 15, 17, 20} Page 27 of 164 MAT101: General Mathematics 1 References 10. Nwagbogwu, D. C. and Akinfenwa, O. A. (2008). Fundamentals of Mathematics, S-S Stephen’s Nig. Ltd., Lagos, Nigeria. 11. www.oneaccess.com.ng/results_by_category.php?CategoryID. 12. Matthews, K. R. (1998). Elementary Linear Algebra, Department of Mathematics, University of Queensland. 13. www.math.fsu.edu/~dli/matthews.pdf 14. Anthony Barcellos, (1992). Calculus and Analytic Geometry, Fifth edition, Volume 1, American River College, Sacramento, California. 15. www.amazon.com ›... › Science & Mathematics › Mathematics › Calculus Page 28 of 164 MAT101: General Mathematics 1 Summary of Study Session 1 In study session one, you have learnt that: 1. A set is a collection of well defined objects, elements or numbers. 2. Set is described by either of the two ways: (a). by tabular form, e.g., A = {1, 2, 3} (b). by set builder form, e.g., B = {x | 2 x 5} 3. If all elements of a set X are members of the set say Y, then X is said to be a subset of Y, i.e., X Y. The set Y is called the superset of X, i.e., Y X. 4. The universal set, denoted by E or U is the set which contains all the possible elements under consideration. 5. A set which has no element is said to be an empty set or null set, denoted by . 6. A B = {x | x A and x B} 7. A B = {x | x A or x B or both} 8. If A B = , then, A and B are disjoint Page 29 of 164 MAT101: General Mathematics 1 Glossary of Terms Set: collection or a class of well defined objects Set theory: the act of putting like things together. Singleton set: a set containing only one element. Universal set: the set that contains all elements. Page 30 of 164 MAT101: General Mathematics 1 Self-Assessment Questions (SAQs) for Study Session 1 Having completed this study session, you can measure how well you have achieved its Learning Outcomes by answering these questions. You can check your answers with the Notes on the Self-Assessment Questions at the end of this Module. SAQ 1.1 (Objective Questions) Select from the following alternative (a) to (d), the correct answer to each question 1. If U = {5, 6, 7, 8}, P = {6, 7} and Q = {6, 8}, then ( P Q )' is (a) {5, 6, 7} (b) {5, 7, 8} (c) {6, 7} (d) {5, 8} 2. The subsets X, Y and Z of a universal set are defined as X = {a, e, m, p}, Y = {a, e, i, o, u} and Z = {l, m, n, o, p, q, r, s, t, u} The elements of X (Y Z ) are (a) {a, e, m, o, p, u} (b) {m, p, o, u} (c) {a, e, m, o, p} (d) {a, e, m, p, u} 3. Given that P = {1, 3, 4} and Q = {2, 3, 4}, then P Q is (a) {1, 2} (b) {2, 3} (c) {2, 4} (d) {3, 4} 4. If the universal set U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, X = {2, 3, 6, 9} and Y = {1, 2, 4, 6, 10}, find ( X Y )'. (a) X 'Y ' (b) ( X Y )' (c) X 'Y ' (d) X Y 5. If X = {1, 2, 3, 4} and Y = {3, 5, 6}, the elements of ( X Y ) X are (a) {1, 2, 3, 4} (b) {3, 5, 6} (c) {3} (d) {1, 2, 4} 6. Given that P = {1, 2, 3, 4}, Q = {5, 6, 7, 8} and R = {1, 2, 3, 2, 4, 2}, it follows that I. P=R n(R) = 3 III. n( P R ) = 4 Which of the statements above is (are) true? (a) I only (b) II only (c) III only (d) I and III only 7. If P = {3, 1, 0, 5}, Q = {2, 3, 8, 1, 4} and R = {7, 6, 5}, then ( P Q ) R is (a) {5} (b) {3, 8} (c) {1, 3, 5, 6, 7} Page 31 of 164 MAT101: General Mathematics 1 SAQ 1.2 (Theory Questions) 1. If P = {1, 2, 3, 4}, Q = {3, 5, 6}, find (a) PQ (b) P Q (c) ( P Q ) Q (d) ( P Q) P 2. If U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, X = {1, 2, 4, 6, 7, 8, 9}, Y = {1, 2, 3, 4, 7, 9}, Z = {2, 3, 4, 7, 9}, find X Y Z '. 3. Given that U = {1, 2, 3, …, 10}, S = {3, 5, 7, 9} and T = {4, 5, 6, 7}, find (i) S – T (ii) SC TC (iii) T SC (iv) ST 4. If A = {x Z | 0 x 15} and B = {x Z | 3 x 10}, find (i) A – B (ii) B – A (iii) A B (iv) A B. Page 32 of 164 MAT101: General Mathematics 1 Study Session 2: The Venn Diagram Introduction The various set operations can be illustrated using diagrams called Venn diagrams. The Venn diagram was initiated by a British mathematician, John Venn (1834-1883). Consequently, the diagrams were named after him. Basically, the universal set is represented by points in and on a rectangle while subsets are represented by points in and on a sphere or circle inside the rectangle. Learning Outcomes for Study Session Two When you have studied this session, you should be able to: 2.1 Identify and define various types of set operations with their applications. 2.2 Illustrate set operations using Venn diagrams. 2.3 Make use of Venn diagrams. 2.4 Relate the solutions in set to real life problems. Page 33 of 164 MAT101: General Mathematics 1 2.1 Set Operations Case Study 2.1 Let A and B be two non-empty sets in the universal set. Represent the following on a Venn diagram. (i) A B (ii) A B (iii) (A B)C (iv) B – A (v) (B – A)C (vi) A B = (vii) A – B (viii) (A – B)C Solution We shall use the shaded portion in our diagram to represent the required region in the Venn diagram. (i) (ii) A B A B AB AB (iii) (iv) A B A B (A B)C B–A (v) (vi) A B A B (B – A)C = A B c A B (vii) A B (viii) AC B A–B Page 34 of 164 MAT101: General Mathematics 1 Case Study 2.2 Let A, B and C be three non-empty sets in the universal set. Represent the following on a Venn diagram. (i) ABC (ii) ABC (iii) (A B C)C (iv) A B CC Solution ABC AB C (ii) A B C (iii) (iv) A B B A C C (A B C)C A B CC 2.1.1 Power Set The set which contains all its possible subsets, including the empty set and the particular set is called the power set. The number of subsets of power set or a non-empty set A is denoted by n{P(A)} = 2 n , where n is the number of elements of the set A, which means that the power set has 2 n subsets. Case Study 2.3 Let A = {a, b, c}. What is the power set of A? Solution Page 35 of 164 MAT101: General Mathematics 1 n First, we get the number of subsets of A, i.e., P(A) = 2. Since A has 3 elements, then the 3 number of subsets is P(A) = 2 = 8, which tells us that A has 8 subsets and they are given as P(A) = { , {a}, {b}, {c}, {a, b}, {a, c}, {b, c}, {a, b, c}} 2.1.2 Algebra of Set We shall consider the laws which are satisfied by set operations. These laws include: 1. Idempotent laws Given any set A, then (i) AA = A (ii) AA = A 2. Commutative laws (i) A B = B A (ii) AB = BA 3. Associative laws: (i) A (B C) = (A B) C (ii) A (B C) = (A B) C 4. Distributive laws If A, B, and C are any three sets, then A (B C) = (A B) (A C) A (B C) = (A B) (A C) 5. Identity laws (i) A = A (ii) A = (iii) AU = U (iv) AU = A 6. Complement laws If A is any set, then, (i) A AC = U (ii) A AC = (iii) (AC)C = A (iv) UC = (v) (A B)C = AC BC 7. De Morgan’s laws If A and B are any two sets, then (i) (A B)C= AC BC (A B)C = AC BC Page 36 of 164 MAT101: General Mathematics 1 Case Study 2.4 The above laws can be proved as follows: 1. (i) Show that A A = A Solution A AA Conversely, let x A A, then x A or x A. Clearly, x A, hence, AAA Therefore, A A = A (ii) Show that A A = A From the definition of the intersection of sets, A A A. Conversely, let x A A. Then, x A and x A. So, x A Hence, A A A. Therefore, A A = A 2. Show that A B = B A Solution Let x A B. Then, x A or x B. This implies that x B or x A. Therefore, x B A So, ABBA (1) Conversely, let y A B. Then, y B or y A. This implies y A or y B. Therefore, y A B Hence, B A A B. (2) By (1) and (2), we have, AB = BA Page 37 of 164 MAT101: General Mathematics 1 3. If A, B, and C are any sets, show that (i) A (B C) = (A B) C (Associative law) Solution Let x A (B C). Then, x A or x B C x B or x C So, x A or x B or x C x A B or x C Therefore, x (A B) C Hence, A (B C) (A B) C (1) Conversely, let y (A B) C, then y A B or y C Thus, y A or y B or y C Therefore, y A (B C) Hence, (A B) C A (B C) (2) By (1) and (2), A (B C) = (A B) C 4. If A, B, and C are any three sets, show that A (B C) = (A B) (A C) Solution Let x A (B C).This implies that x A and x B C Now, x B C implies x B or x C or x both B and C. If x B, then x A B If x C, then x A C This implies x A B or x A C That is, x (A C) (A C) So, A (B C) (A B) (A C) (1) Conversely, x (A B) (A C) This implies that x A B or x A C That is, x A and x B or x A and x C Page 38 of 164 MAT101: General Mathematics 1 This implies x A and x B or x C which implies x A (B C) Hence, (A B) (A C) A (B C) (2) By (1) and (2), A (B C) = (A B) (A C) 5. If A and B are two sets, show that (A B)C = AC BC (De Morgan’s law) Solution Let x (A B)C. Then, by definition of complement, x A B implies x A or x B which implies x A and x B This implies x AC and x BC Thus, x AC BC Hence, (A B)C AC BC (1) Conversely, let x AC BC, then by definition of intersection, x AC and x BC. This implies x A and x B Thus, x A or x B Which implies x A B This implies x (A B)C Therefore, AC BC (A B)C (2) By (1) and (2), (A B)C = AC BC 2.1.3 Cardinality of Sets The cardinality of a set A is the number of elements in the set. It is often denoted by n (A). Page 39 of 164 MAT101: General Mathematics 1 Case Study 2.5 Let A = {a, b, 1, 2, 3}. What is the cardinality of A? Solution Since the cardinality is the number of elements in A, we count the elements and we have, n (A) = 5 Example 2.6 Let B = {3, 5, 7}. What is the cardinality of B? Solution n (B) = 3 2.2 Application of Set theory To this point we have learnt the fundamentals of set theory. Now we shall study the interpretation of each concept in real life situation. This is rather achieved by considering practical examples. Case Study 2.7 Use Venn diagram to represent the following sets: 𝒰 = {1, 3, 5, 7, 11, x, y, z} A = {9, 11, 3, x, y} B = {5, 7, 9, y, z} C = {1, 7, 11, x, y, z} Solution Observe that A 𝒰, B 𝒰 and C 𝒰. So we have, Page 40 of 164 MAT101: General Mathematics 1 Case Study 2.8 If n (X) = 30, n (Y) = 8 and n (X Y) = 38, find n (X Y) and show it on a Venn diagram. Solution n (X Y) = n (X) + n (Y) – n (X Y) n (X Y) = n (X) + n (Y) - n (X Y) = 30 + 8 – 38 =0 Case Study 2.9 Out of 500 health workers investigated, 400 had HIV and 220 had malaria; 70 had both HIV and malaria. Is this data correct? Solution Let U be the set of all health workers that were investigated. Let A be the set of those workers who had HIV. Let B be the set of those workers who had malaria. Thus, A B is the set of workers who had both HIV and malaria. Then, n (U) = 500, n (A) = 400, n (B) = 220 and n (A B) = 70 Therefore, n (A B) = n (A) – n (B) + n (A B) = 400 + 220 – 70 Since we were not told that some workers had neither HIV nor malaria, then, AC BC = or n (AC BC) = 0 And since n (U) = n (A B) + n (A B)C = n (A B) + n (AC BC) = n (A B) + 0 Page 41 of 164 MAT101: General Mathematics 1 But 500 550 and since this value exceeds the total number of workers investigated, the given data is not correct. Case Study 2.10 In the Ministry of Finance, there are 300 employees, out of which 180 are men, 176 are university graduates, 167 are married persons, 84 are male university graduates, 115 are married university graduates, 69 are married men, 27 are married male university graduates. Find the number of single women who are not university graduates. Solution Let 𝒰 be the set of employees; A, the set of men; B, the set of married persons; C, the set of university graduates Then, A B is the set of married men A C is the set of male university graduates B C is the set of married university graduates A B C is the set of married male university graduates. Now, n (𝒰) = 300, n (A) = 180, n (B) = 167, n (C) = 176, n (A B) = 69, n (A C) = 84, n (B C) = 115, n (A B C) = 27. n (AC BC CC) = ? This information can be represented in a Venn diagram as shown below: n (A B C) AC = female BC = single persons CC = not a university graduate Page 42 of 164 MAT101: General Mathematics 1 Then, (i) n (A B) = n (A B CC) + n (A B C) n (A B CC) = n (A B) – n (A B C) = 69 – 27 = 22 which is the number of married male employees who are not university graduates (ii) n (B C) = n (B C AC) + n (A B C) n (B C AC) = n (B C) – n (A B C) = 115 – 27 = 88 This gives the number of married female university graduates. (iii) n (A C) = n (A C BC) + n (A B C) n (A C BC) = n (A C) – n (A B C) = 84 – 27 = 57 This gives the number of male single university graduates. (iv) n (A) = n (A BC CC) + n (A B CC) + n (A BC C) + n (A B C) 180 = n (A BC CC) + 42 + 57 + 27 n (A BC CC) = 180 – 126 = 54 This gives the number of male, single and non-university graduates. We are expected to find the number of single women who are not university graduates. That is, n (A BC CC) Now, n (𝒰) = n (A B C) + n (A B C)C = n (A B C) + n (AC BC CC) (De Morgan’s law) But n (A B C) = n (A) + n (B) + n (C) – n (A B) – n (A C) – n (B C) + n (A B C) = 180 + 167 + 176 – 69 – 84 – 115 + 27 = 282 So, n (𝒰) = 282 + n (AC BC CC) n (AC BC CC) = n (𝒰) – 282 = 300 – 282 = 18 Therefore, the number of single women who are not university graduates is 18. Page 43 of 164 MAT101: General Mathematics 1 Case Study 2.11 In a business class containing 40 students, a student can either take Economics or Accounting or both. If 20 students take Economics, 26 students take Accounting and 4 do not take either subject, find (i) how many take both Economics and Accounting (ii) how many take Economics only. Solution Let the students who take both be X. Let 𝒰 be the set of all students in the class; E, the set of all students who take Economics; A, the set of all students who take Accounting Then, n (𝒰) = 40, n (E) = 20, n (A) = 26, n (E A)C = 4 Representing this information on a Venn diagram, we have, Now, n (𝒰) = n (E A) + n (E A)C = n (E) + n (A) – n (E A) + n (E A)C 40 = 20 - x+ 26 –x +4+x = 50 - x Therefore, x = 50 – 40 = 10 students (ii) Economics only means those that take Economics but not Accounting. That is, n (E AC) = n (E) – n (E A) = 20 – X = 20 – 10 = 10 students Now that you have read the examples above, you can attempt the following In-Text Question (ITQ) Page 44 of 164 MAT101: General Mathematics 1 In a survey of 200 workers, 130 drink Coca-cola and 100 drink Fanta. How many workers drink both Coca-cola and Fanta if only 5 workers drink neither Coca-cola nor Fanta. How many workers drink at least one of Coca-cola and Fanta. Let C represent Coca-cola and F represent Fanta. Let X represent workers who drink both Coca-cola and Fanta. Then, n (U) = 200, n (C) = 130, n (F) = 100 and n (C F)C= 5. Representing this information on a Venn diagram, we have, n (C F)C = 5 (i) n (𝒰) = n (C F) + n (C F)C = n (C) + n (F) – n (C F) + n (C F)C = 130 + 100 – x + 5 x = 235 – 200= 35 So, 35 workers drank both Coca-cola and Fanta. (ii) At least one of Coca-cola or Fanta means either the workers drank Coca-cola but not Fanta or Fanta, but not Coca-cola or both. That is, we want to find n (C FC) + n (CC F) + n (C F)= 130 – x + 100 – x +x= 130 – 35 + 100 – 35 + 5 = 195 Alternatively: The statement: At least one of Coca-cola or Fanta means all the workers that drank Coca-cola or Fanta. That is C F. n (C F) = n (C) + n (F) – n (C F) = 130 + 100 – x = 130 + 100 – 35 = 195 Page 45 of 164 MAT101: General Mathematics 1 Summary of Study Session 2 In study session two, you have learnt that: 1. The set of all the subsets of a set X is called the power set of X, denoted by P(X) 2. If a set has n elements, the number of subsets is in the power set of X is 2 n 3. Venn diagram is a pictorial representation of set 4. n (A B) = n (A) + n (B) – n (A B) 5. n (A B C) = n (A) + n (B) + n (C) – n (A B) – n (A C) – n (B C) + n (A B C) Page 46 of 164 MAT101: General Mathematics 1 Self-Assessment Questions (SAQs) for Study Session 2 Having completed this study session, you can measure how well you have achieved its Learning Outcomes by answering these questions. You can check your answers with the Notes on the Self-Assessment Questions at the end of this Module. SAQ 2.1 A panel of 10 interviewers was to interview two candidates A and B to decide who was suitable for a job. 7 said A was suitable, 5 said B was suitable while 2 said neither A nor B was suitable. (i) How many said both A and B were suitable. (ii) How many said A alone was suitable. In a survey of 200 housewives, it was discovered that 8 had read magazine A, 100 had read magazine B and 96 had read magazine C. It was further discovered that 24 had read A and B, 36 had read B and C while 26 had read A and C. Find i. The number of housewives that had read all three magazines. ii. The number of housewives that had read at least two magazines. iii. The number of housewives that had read only one magazine. SAQ 2.2 In a class of 50 students for a second semester examination, 30 students offer Mathematics, 23 offer Biology while 15 offer Physics. 10 offer Mathematics and Biology, 5 offer Biology and Physics and 6 offer Mathematics and Physics. 2 students do not offer any of the three subjects. i. Draw the Venn diagram to illustrate this information. ii. How many students offer all three subjects? iii. How many students offer any combination of two subjects only? In a class of 40 students, 25 speak Hausa, 16 speak Igbo, 21 speak Yoruba and each of the students speaks at least one of these languages. If 8 speak Hausa and Igbo, 11 speak Hausa and Yoruba and 6 speak Igbo and Yoruba, i. draw a Venn diagram to illustrate this information. Page 47 of 164 MAT101: General Mathematics 1 ii. how many students speak all the three languages? In a class of 36 students, 29 study Mathematics and 20 study Chemistry. If 5 students do neither, how many students study Chemistry but not Mathematics? Page 48 of 164 MAT101: General Mathematics 1 Study Session 3: Number System Introduction The concept of putting more than one objects together brings about numbering. A father might tell his son to bring him oranges and the son ask how many, the father displays 3 fingers, the son says ok and enters inside to bring 3 oranges because he already knew the his father ask him to bring 3 oranges. This concept was developed in different levels till it got to counting in 1,2,3,4,5.. that we are used to this modern time. In this study session, you will be introduced to the properties of natural numbers on addition and multiplication, the meaning of rational, irrational numbers and integers. Learning Outcomes for Study 3 At the end of this session, you should be able to: 3.1 Understand the real number system 3.2 Explain the properties of natural numbers Page 49 of 164 MAT101: General Mathematics 1 3.1 The Real Number System Mathematics began with counting and numbering. Human beings have always had the reason to count because things appears mostly in multiples. The early men put together stones and sticks to denote the figure of each objects. There have also been many elements of counting since then which involves the Hindu-Arabic numeral system, the unary numeral system, the ancient Egyptian numeral system, the Roman numeral system and the arithmetic numerals among others. A number system is a writing system for expressing numbers of a given set using digits or other symbols in a consistent manner. The arithmetic numerals which involves counting numbers in 1,2,3,4,5…. is the most frequently used. These made up the set of natural numbers, which is denoted by N. In-text Questions (ITQs) 3.1 i. What is a number system ii. The most frequently used number system is ___________ In-text Answer (ITAs) 3.1 i. A number system is a writing system for expressing numbers of a given set using digits or other symbols in a consistent manner. ii. Arithmetic numerals Page 50 of 164 MAT101: General Mathematics 1 3.2 Properties of Natural Numbers N The set N of natural numbers has the following properties: i. 𝐼 𝜖 𝑁ie N is a non empty set. ii. For each n 𝜖 𝑁, there exist (n+1) called the successor n. It then follows that (m+1) = (n+1) if and only if m = n. 3.2.1 Addition For any n 𝜖 𝑁, if 𝑚∗ is the successor of m as defined above, then 𝑚∗ + n is defined as (𝑚 + 𝑛)∗. That is, the addition of any two numbers gives rise to another number. 3.2.2 Multiplication (.) For any n 𝜖 𝑁, n.1 is n. That is, any number multiplied by 1 gives us that exact number. For 𝑚, 𝑛 𝜖 𝑁, 𝑚. 𝑛∗ = 𝑚𝑛 + 𝑚, where 𝑛∗ is the successor of n. Case Study 3.1 Let m=2, n=3, 𝑛∗ =4, 𝑚. 𝑛∗ = 2×4 = 8 m.n+m = 2×3+2 = 6+2 = 8 1.5 Order ‘>’ Let m, n𝜖 N, We say m is greater than n (m>n) or n is less than m (nb. or a=b or b>a. In-text Questions (ITQs) 3.2 i. What are the properties of addition and Multiplication ii. What is associative law. In-text Answers (ITAs) 3.2 i. Associative law, Commutative law and Distributive law ii. (a+b)+c= a+(b+c) 3.2.9 Integer The set of natural number N can be extended or include the solution of equation of the type a+x=b with 𝑎, 𝑏𝜖𝑁. We then have the numbers …-3,-2,-1,0,1,2,3… The set consists of these numbers is called the set Z of integers. 3.2.10 Rational Numbers 𝑎 This is the set of all numbers that can be expressed in the form 𝑏, where a≠0 and a,b are 1 52 integers.. Example, 5, 6,8 etc. 3.2.11 Irrational Numbers These are real numbers that are not rational. Example √2, √3, 𝜋 3.2.12 Operations on Real Numbers There are series of laid down rules which governs the multiplication and addition of real numbers. These rules are: 3.2.13 Operation Involving Zero i. The product of any number with zero is zero. Example 5×0=0 ii. The addition of any number to zero is that same number. Example 5+0=5 Page 54 of 164 MAT101: General Mathematics 1 3.2.14 Multiplication by ‘-1’ The following rules applies to any number multiplied by -1 i. (-1)x = -x. Example (-1)5 = -5. ii. -1(-x) = x. Example -1(-5) = 5. iii. (-1)(-x) = x. Example (-1)(-5) = 5. iv. (-x)(-y) = xy. Example (-2)(-5) = 10 v. (-x)y = x(-y) = -xy. Example (-2)5 = 2(-5) = -10. vi. –(x+y) = -x-y. Example –(2+5) = -2-5 = -7 vii. –(x-y) = -x+y. Example –(2+5) = -2+5 = +3. 3.2.15 Operations involving -1 as Index i. −1−1 = −1 1 1 ii. For x≠0, 𝑥 −1 = 𝑥. Example 2−1 = 2 iii. For x≠0, (𝑥 −1 )−1 = 𝑥. Example (2−1 )−1 = 2−1×−1 = 21 = 2 1 1 1 1 1 1 iv. For x≠0 and y≠0, 𝑥𝑦 −1 = 𝑥 −1 × 𝑦 −1 , 𝑥 × 𝑦. 𝐸𝑥𝑎𝑚𝑝𝑙𝑒 =2×3= 2×3 6 𝑥 𝑦 2 3 v. For x≠0 and y≠0, (𝑦)−1 = 𝑥. 𝐸𝑥𝑎𝑚𝑝𝑙𝑒(3)−1 = 2. Page 55 of 164 MAT101: General Mathematics 1 Figure 1.1: Properties of numbers Page 56 of 164 MAT101: General Mathematics 1 In Summary of Study Session 3, you have learnt that: i. A number system is a writing system for expressing numbers of a given set using digits or other symbols in a consistent manner. ii. The addition of any two numbers gives rise to another number. iii. Any number multiplied by 1 gives us that exact number. iv. Addition is associative and commutative v. Multiplication is associative and commutative vi. Distributive of multiplication is over addition vii. The set consists of natural numbers is called the set Z of integers. viii. The product of any number with zero is zero. Example 5×0=0 ix. The addition of any number to zero is that same number. Example 5+0=5 Page 57 of 164 MAT101: General Mathematics 1 Self-Assessment Questions(SAQs) Now that you have completed this study session, you can access how well you have achieved its Learning outcomes by answering these questions. Write your answers in your diary and discuss them with your tutor at the next support meeting. You can check your answers with the notes on the self-assessment questions at the end of this module. SAQ 3.1 Explain the number system SAQ 3.2 If a=11, b=23, show that the addition of these numbers give rise to another number. Show that any number multipled by 1 gives us that same number. What are the properties of natural numbers? If a=5, b=7 and c=13, show all the properties of natural numbers.. Page 58 of 164 MAT101: General Mathematics 1 References 1. Usman, M.A., Odetunde, O.S., Ogunwobi, Z.O., Hammed, F.A.,(2016). Mathematics for University Students, Vol 1. Ibadan. Page 59 of 164 MAT101: General Mathematics 1 Study Session 4: Mathematical Induction Introduction Before a theory is generally accepted scientifically, it has to go through a lot of proofs. One of the essential tools of proving in mathematics ismathematical induction; it is a technique for proving results or establishing statements for natural numbers. This study session illustrates the method through a variety of examples. Learning Outcomes for Study Session 4 When you have studied this session, you should be able to: 4.1 Define mathematical induction 4.2 Use mathematical induction to prove statement for natural numbers Page 60 of 164 MAT101: General Mathematics 1 4.1 Definition Mathematical Induction is a mathematical technique which is used to prove a statement, a formula or a theorem is true for every natural number. The technique involves two steps to prove a statement, as stated below: Step 1(Base step): It proves that a statement is true for the initial value. Step 2(Inductive step): It proves that if the statement is true for the nth iteration (or number n), then it is also true for (𝑛 + 1)𝑡ℎ h iteration ( or number n+1). In-Text Question (ITQ) 4.1 i. Define mathematical induction In-Text Answer (ITA) 4.1 i. Mathematical Induction is a mathematical technique which is used to prove a statement, a formula or a theorem is true for every natural number 4.2 Proving Statements with Mathematical Induction Step 1: Consider an initial value for which the statement is true. It is to be shown that the statement is true for n = initial value. Step 2: Assume the statement is true for any value of n = k. Then prove the statement is true for n = k+1. We actually break n = k+1 into two parts, one part is n = k (which is already proved) and try to prove the other part. Case Study4.1 Prove 3𝑛 − 1 is a multiple of 2 for n = 1, 2,... Solution Step 1: For n = 1, 31 − 1 = 3 − 1 = 2 which is a multiple of 2 Page 61 of 164 MAT101: General Mathematics 1 𝑛 𝑘 Step 2: Let us assume 3 − 1 is true for n=k, Hence, 3 − 1 is true (It is an assumption) We have to prove that 3𝑘+1 − 1 is also a multiple of 2 3𝑘+1 − 1=3 × 3𝑘+1 − 1=(2 × 3𝑘 ) = (3𝑘 − 1) The first part (2 × 3𝑘 ) is certain to be a multiple of 2 and the second part (3𝑘 − 1) is also true as our previous assumption. Hence, 3𝑘+1 − 1is a multiple of 2. So, it is proved that 3𝑛 − 1 is a multiple of 2. Case Study 4.2 1 + 3 + 5+... +(2𝑛 − 1) = 𝑛2 𝑓𝑜𝑟 𝑛 = 1,2, … Solution Step 1: For n=1,1=12 , Hence, step 1 is satisfied. Step 2: Let us assume the statement is true for n =k Hence, 1 + 3 + 5 + ⋯ + (2𝑘 − 1) = 𝑘 2 is true (It is an assumption) We have to prove that 1 + 3 + 5+... +(2(𝑘 + 1) − 1) = (𝑘 + 1)2 also holds 1+3 + 5 + ⋯ + (2(𝑘 + 1) − 1) = 1 + 3 + 5 + ⋯ + (2𝑘 + 2 − 1) =1+3+5+⋯+(2k+1) =1+3+5+⋯+(2k−1)+(2k+1) = 𝑘 2 + (2𝑘 + 1) Page 62 of 164 MAT101: General Mathematics 1 2 = (𝑘 + 1) 𝑆𝑜, 1 + 3 + 5 + ⋯ + (2(𝑘 + 1) − 1) = (𝑘 + 1)2 hold which satisfies the step 2. Hence, 1 + 3 + 5 + ⋯ + (2𝑛 − 1) = 𝑛2 is proved. Case Study 4.3 Prove that (𝑎𝑏)𝑛 = 𝑎𝑛 𝑏 𝑛 is true for every natural number n Solution Step 1: For n=1, (𝑎𝑏)1 = 𝑎1 𝑏1 = 𝑎𝑏, Hence, step 1 is satisfied. Step 2: Let us assume the statement is true for n=k , Hence, (𝑎𝑏)𝑘 = 𝑎𝑘 𝑏 𝑘 is true (It is an assumption). We have to prove that (𝑎𝑏)𝑘+1 = 𝑎𝑘+1 𝑏 𝑘+1 also hold Given, (𝑎𝑏)𝑘 = 𝑎𝑘 𝑏 𝑘 Or, (𝑎𝑏)𝑘 (𝑎𝑏) = 𝑎𝑘 𝑏 𝑘 (𝑎𝑏) [Multiplying both side by 'ab'] Or, (𝑎𝑏)𝑘+1 = (𝑎𝑎𝑘 )(𝑏𝑏𝑘 ) Or, (𝑎𝑏)𝑘+1 =(𝑎𝑘+1 𝑏 𝑘+1 ) Hence, step 2 is proved. So, (𝑎𝑏)𝑛 = 𝑎𝑛 𝑏 𝑛 is true for every natural number n. 4.2.1 Strong Induction Strong Induction is another form of mathematical induction. Through this induction technique, we can prove that a propositional function, P(n) is true for all positive integers, n, using the following steps: Page 63 of 164 MAT101: General Mathematics 1 a. Step 1(Base step): It proves that the initial proposition P(1) is true. b. Step 2(Inductive step): It proves that the conditional statement [P(1) ∧P(2) ∧P(3) ∧⋯∧P(k)] → P(k+1) is true for positive integers k Page 64 of 164 MAT101: General Mathematics 1 Summary In this study session, you have learnt that: i. Mathematical Induction is a mathematical technique which is used to prove a statement, a formula or a theorem is true for every natural number ii. The technique involves two steps to prove a statement are base step and inductive step iii. Base step proves that a statement is true for the initial value. iv. Inductive step proves that if the statement is true for the nth iteration (or number n), then it is also true for (𝑛 + 1)𝑡ℎ h iteration ( or number n+1). Page 65 of 164 MAT101: General Mathematics 1 Self-Assessment Questions (SAQs) Now that you have completed this study session, you can access how well you have achieved its Learning outcomes by answering these questions. Write your answers in your diary and discuss them with your tutor at the next support meeting. You can check your answers with the notes on the self-assessment questions at the end of this module. 1 i. Using mathematical induction, prove that 12 + 22 + 32 … + 𝑛2 = (6) {𝑛(𝑛 + 1)(2𝑛 + 1)} 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑛 ∈ 𝑁 ii. By mathematical induction, prove that 1 × 2 + 3 × 4 + 5 × 6 + ⋯ + (2𝑛 − 1) × 𝑛(𝑛+1)(4𝑛−1) 2𝑛 = 3 Page 66 of 164 MAT101: General Mathematics 1 Study Session 5: Real Sequences and Series Introduction Consider each of the following set of numbers: a. 1, 3, 5, 7, 9... b. 20, 17, 14, 11... c. 3, 6, 12, 24, 48... d. 1, 4, 9, 16,... These are examples of a sequence. Every member of each set is called a term, the three dots after each set show that the set of numbers continues indefinitely. A sequence is an arrangement of a set of numbers in a particular order followed by some rules.To know the nth term of a sequence, it is important to study the rule guiding its increment or decrement by checking the first 2 terms, apply the rule to the next term. For instance, in (a) above, each term is 2 more than the preceding term, so the sequence continues as …11, 13, 15…, in (b) above, each term is 3 less than the preceding terms, which means you will subtract 3 from each term to generate the next term, so the sequence continues as …8, 5, 2… In (c) above, each term is multiplied by 2 to get the next term and the sequence continues as …96, 192, 384… So also (d) above is a set of square numbers i.e. 12 , 22 , 32 , 42 …, so the next terms are 52 , 62 , 72 which is …25, 36, 49… Learning Outcomes for Study Session 5 When you have studied this session, you should be able to: 5.4 Understand what a sequence means 5.5 Calculate the arithmetic progression and arithmetic mean of a sequence 5.6 Solve the geometric progression and geometric mean of a sequence 5.7 Find the sum of arithmetic progression and geometric progression of a sequence. Page 67 of 164 MAT101: General Mathematics 1 5.1 Sequence as a Function Consider a sequence 2, 5, 8, 11, …, the general term is 2 + (𝑛 − 1) × 3 = 2 + 3𝑛 − 3 = 3𝑛 − 1. You can verify by taking n = 1, 2, 3, 4. Find the 10th term (n=10), the 25th term and the 100th term. Hence, it denotes any term T is a function of the positive integers where 𝑇(𝑛) = 3𝑛 − 1. We shall write this as 𝑇𝑛 = 3𝑛 − 1, where 𝑇𝑛 is the nth term and n=1, 2, 3, … This will give you a formula for finding any term of the sequence. In general, a sequence 𝑇1 , 𝑇2 , 𝑇3 , … 𝑇𝑛 is the set of images given by the function 𝑇𝑛 = 𝑓(𝑛) of the positive integers 1, 2, 3… ,n. The sequence may be finite or infinite. In a finite sequence, n will have an upper value and 𝑇𝑛 will be the last term of the sequence while an infinite sequence has no last term. Case Study 5.1 Find the first 5 terms of the sequence whose general term is given by: a. 𝑇𝑛 = 3 + 7𝑛 b. 𝑇𝑛 = (−2)𝑛 Solution a. 𝑇𝑛 = 3 + 7𝑛 𝑇1 = 3 + 7(1) = 10 𝑇2 = 3 + 7(2) = 17 𝑇3 = 3 + 7(3) = 24 𝑇4 = 3 + 7(4) = 31 𝑇5 = 3 + 7(5) = 38 b. 𝑇𝑛 = (−2)𝑛 𝑇1 = (−2)1 = −2 𝑇2 = (−2)2 = 4 𝑇3 = (−2)3 = −8 𝑇4 = (−2)4 = 16 Page 68 of 164 MAT101: General Mathematics 1 5 𝑇5 = (−2) = −32 5.2 Arithmetic Sequence 5.2.1 Arithmetic Progression (A.P.) In an arithmetic progression (Linear sequence), the difference between a term and the one preceding it is always a constant. The constant number is called the common difference denoted by d i.e. 𝑇𝑛 − 𝑇𝑛−1 = 𝑑. Where 𝑇𝑛−1 𝑖𝑠 𝑡ℎ𝑒 (𝑛 − 1)𝑡ℎ 𝑡𝑒𝑟𝑚. Case Study 5.2 Arithmetic Progression (A.P.) Common difference (d) 3, 7, 11, 15… 4 10, 6, 2, -2… -4 -21, -16, -11, -9 5 If a sequence 𝑇1 , 𝑇2 , 𝑇3 , 𝑇4 , is such that 𝑇2 − 𝑇1 = 𝑇3 − 𝑇2 = 𝑇4 − 𝑇3 = 𝑇5 − 𝑇4 … 𝑡ℎ𝑒𝑛, 𝑇1 , 𝑇2 , 𝑇3 , 𝑇4 … 𝑎𝑟𝑒 𝑖𝑛 𝑎𝑟𝑖𝑡ℎ𝑚𝑒𝑡𝑖𝑐 𝑝𝑟𝑜𝑔𝑟𝑒𝑠𝑠𝑖𝑜𝑛. So, for any A.P., 𝑇𝑛 = 𝑎 + (𝑛 − 1)𝑑 Where a is the first term, n is the nth term and d is the common difference. Case Study5.3 What is the 15th term of the sequence -3, 2, 7, … Solution Page 69 of 164 MAT101: General Mathematics 1 d= 2-(-3) = 5, a = -3 𝑇𝑛 = 𝑎 + (𝑛 − 1)𝑑 𝑇15 = −3 + (15 − 1)5 𝑇15 = −3 + (14)5 = −3 + 70 𝑇15 = 67 Case Study 5.4 The 4th term of an arithmetic progression is 15 and the 9th term is 35. Find the 15th term. Solution 𝑇4 = 15, 𝑇9 = 35 𝑇4 = 𝑎 + 3𝑑 = 15 𝑇9 = 𝑎 + 8𝑑 = 35 ∴ 𝑎 + 3𝑑 = 15 (i) 𝑎 + 8𝑑 = 35 (ii) Subtracting (i) from (ii) 5𝑑 = 20 ∴𝑑=4 Substitute 𝑑 = 4 in equation (i) 𝑎 + 3(4) = 15 𝑎 = 15 − 12 𝑎=3 Page 70 of 164 MAT101: General Mathematics 1 𝑇15 = 𝑎 + (15 − 1)𝑑 𝑇15 = 3 + 14 × 4 𝑇15 = 59 5.2.2 Arithmetic Mean If a, b, c are three consecutive terms of an A.P., then b is the arithmetic mean of a and c. 1 Now b – a = c – b, so b = (𝑎 + 𝑐) 2 Case Study 5.5 Find the arithmetic mean of 4 and 18. Solution Let the arithmetic mean be 𝑝, ∴ 4, 𝑝 𝑎𝑛𝑑 18 is an arithmetic progression. 4 + 18 22 ∴𝑝= = 2 2 𝑝 = 11 Case Study 5.6: Insert 5 arithmetic means between -3 and 21 Solution You are simply asked to find 5 numbers p, q, r, s, t, such that -3, p, q, r, s, t, 21 are seven terms of an A.P. Then, 𝑎 = −3, 𝑇7 = 21 So 𝑇7 = −3 + (7 − 1)𝑑 = 21 Page 71 of 164 MAT101: General Mathematics 1 𝑇7 = −3 + (6)𝑑 = 21 𝑇7 = 6𝑑 = 24; 𝑑 = 4 Since d = 4, You will add 4 to the first term and the subsequent terms to have -3, 1, 5, 9, 13, 17, Therefore, the required numbers are 1, 5, 9, 13, 17 5.3 Geometric Sequence In this section, you will learn about geometric progression and arithmetic mean. 5.3.1 Geometric Progression (G.P.) In a geometric progression (Exponential sequence), the ratio of a term and that immediately preceding it is always a constant. This constant number is called the common ration denoted by r. 𝑇 𝑟 = 𝑇 𝑛 , where 𝑇𝑛−1 is the (n - 1)th term. 𝑛−1 𝑇2 𝑇 𝑇 So, if a sequence 𝑇1 , 𝑇2 , 𝑇3 , 𝑇4 … 𝑖𝑠 𝑠𝑢𝑐ℎ 𝑡ℎ𝑎𝑡 = 𝑇3 = 𝑇4…, then 𝑇1 , 𝑇2 , 𝑇3 , 𝑇4 … are in 𝑇1 3 3 geometric progression. Page 72 of 164 MAT101: General Mathematics 1 Case Study 5.7 Geometric Progression (G.P.) Common Ratio 2, 4, 8, 16… 4 8 16 𝑟= = = =2 2 4 8 1 5 1 5 12, 8, 5 , 3 … 8 5 3 2 3 9 𝑟= = 3 = 19 = 12 8 5 3 3 4, -8, 16, -32… −8 16 −32 𝑟= = = = −2 4 −8 16 If the first term of a geometric sequence is a and the common ratio is r, if 𝑇𝑛 is the nth term of the sequence then: 𝑇1 = 𝑎 𝑇2 =𝑟 𝑇1 ∴ 𝑇2 = 𝑇1 × 𝑟 = 𝑎𝑟 𝑇3 = 𝑇2 × 𝑟 = 𝑎𝑟 × 𝑟 = 𝑎𝑟 2 𝑇4 =𝑟 𝑇3 𝑇4 = 𝑇3 × 𝑟 = 𝑎𝑟 2 × 𝑟 = 𝑇𝑛 Thus, the nth term of a geometric progression 𝑇𝑛 is 𝑇𝑛 = 𝑎𝑟 𝑛−1 Where a is the first term and r is the common ratio. Case Study 5.8 The 2nd term of a g.p. is 35 and the fourth term is 875, find: Page 73 of 164 MAT101: General Mathematics 1 a. The first term b. The fifth term Solution
