2712 Precalc SE Tpg.indd 12712 Precalc SE Tpg.indd 1 1/3/05 12:40:36 PM1/3/05 12:40:36 PM

Copyright 2006 by Holt, Rinehart and Winston

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Acknowledgment: Portions of this text were previously published inContemporary Precalculus by Thomas Hungerford, 2000, SaundersPublishing Co., and appear here with permission of the publisher.

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ISBN: 0-03-041647-7

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VV

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V

AUTHORS

VI

Thomas W. Hungerford

Dr. Hungerford, a leading authority in the use of technology inadvanced mathematics instruction, was Professor of Mathematics atCleveland State University for many years. In addition to publishingnumerous research articles, he has authored thirteen mathematicstextbooks, ranging from the high school to the graduate level. Dr. Hungerford was one of the founders of the ClevelandCollaborative for Mathematics Education, a long-term projectinvolving local universities, businesses, and mathematics teachers.

Irene Sam Jovell

An award winning teacher at Niskayuna High School, Niskayuna,New York, Ms. Jovell served on the writing team for the New YorkState Mathematics, Science, and Technology Framework. A popular pre-senter at state and national conferences, her workshops focus ontechnology-based innovative education. Ms. Jovell has served aspresident of the New York State Mathematics Teachers Association.

Betty Mayberry

Ms. Mayberry is the mathematics department chair at Pope John Paul II High School, Hendersonville, Tennessee. She has received thePresidential Award for Excellence in Teaching Mathematics and theTandy Technology Scholar award. She is a Teachers Teaching withTechnology instructor, is a popular speaker for the effective use oftechnology in mathematics instruction, and has served as president of the Tennessee Mathematics Teachers Association and Council ofPresidential Awardees in Mathematics.

Martin Engelbrecht

A mathematics teacher at Culver Academies, Culver, Indiana, Mr.Engelbrecht also teaches statistics at Purdue University, NorthCentral. An innovative teacher and writer, he integrates appliedmathematics with technology to make mathematics accessible to allstudents.

CONTENTCONSULTANT

Preface VII

REVIEWERS

VII

J. AltonjyMontville High School Montville, NJ

Mark BudahlMitchell Public SchoolsMitchell, SD

Ronda DavisSandia High SchoolAlbuquerque, NM

Renetta F. DeremerHollidaysburg Area Senior High SchoolHollidaysburg, PA

James M. HarringtonOmaha Public SchoolsOmaha, NE

Mary MeierottoCentral High SchoolDuluth, MN

Anita MorrisAnn Arundel County Public SchoolsAnnapolis, MD

Raymond Scacalossi Jr.Hauppauge SchoolsHauppauge, NY

Harry SirockmanCentral Catholic High SchoolPittsburgh, PA

Marilyn WislerHazelwood West High SchoolHazelwood, MO

Cathleen M. Zucco-TeveloffTrinity CollegeHartford, CT

Charlie BialowasAnaheim Union High SchoolAnaheim, CA

Marilyn CobbLake Travis High SchoolAustin, TX

Jan DeibertEdison High SchoolHuntington Beach, CA

Richard F. DubeTaunton Public SchoolsTaunton, MA

Jane La VoieGreece Arcadia High SchoolRochester, NY

Cheryl MockelMt. Spokane High SchoolMead, WA

Joseph NidyMayfield High SchoolMayfield Village, OH

Eli ShaheenPlum Senior High Pittsburgh, PA

Catherine S. WoodChester High SchoolChester, PA

Janie ZimmerResearch For Better SchoolsPhiladelphia, PA

VIII PrefaceVIII PrefaceVIII Preface

PREFACEThis book is intended to provide the mathematical background need-ed for calculus, and it assumes that students have taken a geometrycourse and two courses in algebra. The text integrates graphing tech-nology into the course without losing the underlying mathematics,which is the crucial issue. Mathematics is presented in an informalmanner that stresses meaningful motivation, careful explanations,and numerous examples, with an ongoing focus on real-world prob-lem solving.

The concepts that play a central role in calculus are explored fromalgebraic, graphical, and numerical perspectives. Students are expect-ed to participate actively in the development of these concepts byusing graphing calculators or computers with suitable software, asdirected in the Graphing Explorations and Calculator Explorations, either to complete a particular discussion or to explore appropriateexamples.

A variety of examples and exercises based on real-world data areincluded in the text. Additionally, sections have been included cover-ing linear, polynomial, exponential, and logarithmic models, whichcan be constructed from data by using the regression capabilities of a calculator.

Chapter 1 begins with a review of basic terminology. Numerical pat-terns are discussed that lead to arithmetic sequences, lines, and linearmodels. Geometric sequences are then introduced. Some of thismaterial may be new to many students.

Chapter 2 introduces solving equations graphically and then reviewstechniques for finding algebraic solutions of various types of equa-tions and inequalities.

Chapter 3 discusses functions in detail and stresses transformations ofparent functions. Function notation is reviewed and used throughoutthe text. The difference quotient, a basic building block of differential

Representations

Organization of Beginning Chapters

Preface IXPreface IX

calculus, is introduced as a rate-of-change function; several examplesare given. There is an optional section on iterative real-valuedfunctions.

Chapter 4 reviews polynomial and rational functions, introduces com-plex numbers, and includes an optional section on the Mandelbrotset. Finally, the Fundamental Theorem of Algebra is introduced.

Chapter 5 reviews and extends topics on exponential and logarithmicfunctions.

Five full chapters offer extensive coverage of trigonometry. Chapter 6introduces trigonometry as ratios in right triangles, expands the dis-cussion to include angle functions, and then presents trigonometricratios as functions of real numbers. The basic trigonometric identitiesare given, and periodicity is discussed.

Chapter 7 introduces graphs of trigonometric functions and discussesamplitude and phase shift.

Chapter 8 deals with solving trigonometric equations by using graph-ical methods, as well as finding algebraic solutions by using inversetrigonometric functions. Algebraic methods for finding solutions totrigonometric equations are also discussed. The last section ofChapter 8 introduces simple harmonic motion and modeling.

Chapter 9 presents methods for proving identities and introducesother trigonometric identities.

Chapter 10 includes the Law of Cosines, the Law of Sines, polar formof complex numbers, de Moives theorem, and nth roots of complexnumbers. Vectors in the plane and applications of vectors are alsopresented.

Chapters 11 through 14 are independent of each other and may bepresented in any order. Topics covered in these chapters include ana-lytic geometry, systems of equations, statistics and probability, andlimits and continuity.

Chapter Openers Each chapter begins with a brief example of anapplication of the mathematics treated in that chapter, together with areference to an appropriate exercise. The opener also lists the titles ofthe sections in the chapter and provides a diagram showing theirinterdependence.

Excursions Each Excursion is a section that extends or supplementsmaterial related to the previous section. Some present topics thatillustrate mathematics developed with the use of technology, someare high-interest topics that are motivational, and some present mate-rial that is used in other areas of mathematics. Exercises are includedat the end of every Excursion. Clearly marked exercises reflectingmaterial contained in each Excursion are also in correspondingChapter Reviews. Each Excursion is independent of the rest of thebook and should be considered an extension or enrichment.

Trigonometry

Organization ofEnding Chapters

Features

X PrefaceX PrefaceX Preface

Cautions Students are alerted to common errors and misconceptions,both mathematical and technological, by clearly marked Cautionboxes.

Notes Students are reminded of review topics, or their attention isdirected toward specific content.

Exercises Exercise sets proceed from routine calculations and drill toexercises requiring more complex thought, including graph interpre-tation and word problems. Problems labeled Critical Thinking presenta question in a form different from what students may have seenbefore; a few of the Critical Thinking problems are quite challenging.Answers for selected problems are given in the back of the book.

Chapter Reviews Each chapter concludes with a list of importantconcepts (referenced by section and page number), a summary ofimportant facts and formulas, and a set of review exercises.

Technology Appendix The technology appendix presents an overviewof the use of the graphing calculator. It is recommended that studentswho are unfamiliar with the use of a graphing calculator complete allexamples, explorations, investigations, and exercises in this appendix.All students may use the appendix for reference.

Algebra Review This Appendix reviews basic algebra. It can be omit-ted by well-prepared students or treated as an introductory chapter.Exercises are included.

Geometry Review Frequently used facts from plane geometry aresummarized, with examples, in this appendix.

Mathematical Induction and the Binomial Theorem Material relevantto these two topics is presented in an appendix with examples andexercises.

Minimal Technology Requirements It is assumed that each studenthas either a computer with appropriate software or a calculator at thelevel of a TI-82 or higher. Among current calculator models that meetor exceed this minimal requirement are TI-82 through TI-92, Sharp9900, HP-39, and Casio 9850 and 9970. All students unfamiliar withgraphing technology should complete the Technology Appendixbefore beginning the material.

Because either a graphing calculator or a computer with graphingsoftware is required, several features are provided in the text to assistthe student in the use of these tools.

Technology Tips Although the discussion of technology in the text isas generic as possible, some Technology Tips provide information andassistance in carrying out various procedures on specific calculators.Other Tips offer general information or helpful advice about perform-ing a particular task on a calculator.

Appendices

Technology

Preface XIPreface XI

To avoid clutter, only a limited number of calculators are specificallymentioned in the Technology Tips. However, unless noted otherwise,observe the following guidelines.

Technology Tips for TI apply to TI-84 Plus, TI-83 Plus, TI-83,andexcept for some matrix operationsTI-82

Technology Tips for TI-86 also apply to TI-85 Technology Tips for Casio apply to Casio 9850GB-Plus, Casio

9850, and Casio 9970

Calculator Explorations Students are directed to use a calculator orcomputer with suitable software to complete a particular discussionor to explore certain examples.

Graphing Explorations Students may not be aware of the full capabilitiesor limitationsof a calculator. The GraphingExplorations will help students to become familiar with the calculator and to maximize mathematical power. Even if the instructor does not assign these investigations, students may want to read through them for enrichment purposes.

Each chapter has a Can Do Calculus feature that connects a calculustopic to material included in that chapter. This feature gives the stu-dent the opportunity to briefly step into the world of calculus. Manyof these features include topics that are typically solved by using cal-culus but can be solved by using precalculus skills that the studenthas recently acquired. Other Can Do Calculus features conceptuallydevelop calculus topics by using tables and graphs.

The chart on the next page shows the interdependence of chapters. Asimilar chart appears at the beginning of each chapter, showing theinterdependence of sections within the chapter.

Can Do CalculusFeatures

Interdependence ofChapters and Sections

INTERDEPENDENCE OF CHAPTERS

XII

4Polynomial and Rational

Functions

9Trigonometric Identities

and Proof

8Solving Trigonometric

Equations

5Exponential and

Logarithmic Functions

11Analytic Geometry

12Systems and Matrices

13Statistics and

Probability

14Limits and Continuity

1Number Patterns

10TrigonometricApplications

3Functions and Graphs

2Equations and

Inequalities

7Trigonometric Graphs

6Trigonometry

TABLE OF CONTENTS

Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .VIIIApplications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . XVIII

C H A P T E R 1 Number Patterns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.1 Real Numbers, Relations, and Functions . . . . . . . . . . . . . . . . . 31.2 Mathematical Patterns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131.3 Arithmetic Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211.4 Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301.5 Linear Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 431.6 Geometric Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

Chapter Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65can do calculus Infinite Geometric Series . . . . . . . . . . . . 76

C H A P T E R 2 Equations and Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . 802.1 Solving Equations Graphically . . . . . . . . . . . . . . . . . . . . . . . 812.2 Soving Quadratic Equations Algebraically . . . . . . . . . . . . . . . 882.3 Applications of Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 972.4 Other Types of Equations . . . . . . . . . . . . . . . . . . . . . . . . . . 1072.5 Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

2.5.A Excursion: Absolute-Value Inequalities . . . . . . . . 127Chapter Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133can do calculus Maximum Area . . . . . . . . . . . . . . . . . . 138

XIII

C H A P T E R 3 Functions and Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1403.1 Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1413.2 Graphs of Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1503.3 Quadratic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1633.4 Graphs and Transformations . . . . . . . . . . . . . . . . . . . . . . . . 172

3.4.A Excursion: Symmetry . . . . . . . . . . . . . . . . . . . . . . 1843.5 Operations on Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 191

3.5.A Excursion: Iterations and Dynamical Systems . . . 1993.6 Inverse Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2043.7 Rates of Change . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214

Chapter Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224can do calculus Instantaneous Rates of Change . . . . . . . 234

C H A P T E R 4 Polynomial and Rational Functions . . . . . . . . . . . . . . . . 2384.1 Polynomial Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2394.2 Real Zeros . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2504.3 Graphs of Polynomial Functions . . . . . . . . . . . . . . . . . . . . . 260

4.3.A Excursion: Polynomial Models . . . . . . . . . . . . . . 2734.4 Rational Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2784.5 Complex Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293

4.5.A Excursion: The Mandelbrot Set . . . . . . . . . . . . . . 3014.6 The Fundamental Theorem of Algebra . . . . . . . . . . . . . . . . 307

Chapter Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315can do calculus Optimization Applications . . . . . . . . . . 322

C H A P T E R 5 Exponential and Logarithmic Functions . . . . . . . . . . . 3265.1 Radicals and Rational Exponents . . . . . . . . . . . . . . . . . . . . . 3275.2 Exponential Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3365.3 Applications of Exponential Functions . . . . . . . . . . . . . . . . 3455.4 Common and Natural Logarithmic Functions . . . . . . . . . . . 3565.5 Properties and Laws of Logarithms . . . . . . . . . . . . . . . . . . . 363

5.5.A Excursion: Logarithmic Functions to Other Bases . . . . . . . . . . . . . . . . . . . . . 370

5.6 Solving Exponential and Logarithmic Equations . . . . . . . . . 3795.7 Exponential, Logarithmic, and Other Models . . . . . . . . . . . . 389

Chapter Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 401can do calculus Tangents to Exponential Functions . . . . 408

XIV

C H A P T E R 6 Trigonometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4126.1 Right-Triangle Trigonometry . . . . . . . . . . . . . . . . . . . . . . . . 4136.2 Trigonometric Applications . . . . . . . . . . . . . . . . . . . . . . . . . 4216.3 Angles and Radian Measure . . . . . . . . . . . . . . . . . . . . . . . . 4336.4 Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 4436.5 Basic Trigonometric Identities . . . . . . . . . . . . . . . . . . . . . . . 454

Chapter Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 462can do calculus Optimization with Trigonometry . . . . . 468

C H A P T E R 7 Trigonometric Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4727.1 Graphs of the Sine, Cosine, and Tangent Functions . . . . . . . 4737.2 Graphs of the Cosecant, Secant, and Cotangent Functions . . 4867.3 Periodic Graphs and Amplitude . . . . . . . . . . . . . . . . . . . . . 4937.4 Periodic Graphs and Phase Shifts . . . . . . . . . . . . . . . . . . . . 501

7.4.A Excursion: Other Trigonometric Graphs . . . . . . . . 510Chapter Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 516can do calculus Approximations with Infinite Series . . . 520

C H A P T E R 8 Solving Trigonometric Equations . . . . . . . . . . . . . . . . . . 5228.1 Graphical Solutions to Trigonometric Equations . . . . . . . . . 5248.2 Inverse Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . 5298.3 Algebraic Solutions of Trigonometric Equations . . . . . . . . . 5388.4 Simple Harmonic Motion and Modeling . . . . . . . . . . . . . . . 547

8.4.A Excursion: Sound Waves . . . . . . . . . . . . . . . . . . . 558Chapter Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 563can do calculus Limits of Trigonometric Functions . . . . 566

C H A P T E R 9 Trigonometric Identities and Proof . . . . . . . . . . . . . . . . . 5709.1 Identities and Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5719.2 Addition and Subtraction Identities . . . . . . . . . . . . . . . . . . . 581

9.2.A Excursion: Lines and Angles . . . . . . . . . . . . . . . . 5899.3 Other Identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5939.4 Using Trigonometric Identities . . . . . . . . . . . . . . . . . . . . . . 602

Chapter Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 610can do calculus Rates of Change in Trigonometry . . . . . 614

XV

C H A P T E R 10 Trigonometric Applications . . . . . . . . . . . . . . . . . . . . . . . . . 61610.1 The Law of Cosines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61710.2 The Law of Sines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62510.3 The Complex Plane and Polar Form for Complex Numbers . 63710.4 DeMoivres Theorem and nth Roots of Complex Numbers . 64410.5 Vectors in the Plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65310.6 Applications of Vectors in the Plane . . . . . . . . . . . . . . . . . . . 661

10.6.A Excursion: The Dot Product . . . . . . . . . . . . . . . . 670Chapter Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 681can do calculus Eulers Formula . . . . . . . . . . . . . . . . . . 688

C H A P T E R 11 Analytic Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69011.1 Ellipses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69211.2 Hyperbolas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70011.3 Parabolas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70911.4 Translations and Rotations of Conics . . . . . . . . . . . . . . . . . . 716

11.4.A Excursion: Rotation of Axes . . . . . . . . . . . . . . . . 72811.5 Polar Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73411.6 Polar Equations of Conics . . . . . . . . . . . . . . . . . . . . . . . . . . 74511.7 Plane Curves and Parametric Equations . . . . . . . . . . . . . . . 754

11.7.A Excursion: Parameterizations of Conic Sections . . . . . . . . . . . . . . . . . 766

Chapter Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 770can do calculus Arc Length of a Polar Graph . . . . . . . . 776

C H A P T E R 12 Systems and Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77812.1 Solving Systems of Equations . . . . . . . . . . . . . . . . . . . . . . . 779

12.1.A Excursion: Graphs in Three Dimensions . . . . . . 79012.2 Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79512.3 Matrix Operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80412.4 Matrix Methods for Square Systems . . . . . . . . . . . . . . . . . . 81412.5 Nonlinear Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 821

12.5.A Excursion: Systems of Inequalities . . . . . . . . . . . 826Chapter Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 834can do calculus Partial Fractions . . . . . . . . . . . . . . . . . . 838

XVI

C H A P T E R 13 Statistics and Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . 84213.1 Basic Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84313.2 Measures of Center and Spread . . . . . . . . . . . . . . . . . . . . . . 85313.3 Basic Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86413.4 Determining Probabilities . . . . . . . . . . . . . . . . . . . . . . . . . . 874

13.4.A Excursion: Binomial Experiments . . . . . . . . . . . 88413.5 Normal Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .889

Chapter Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 898can do calculus Area Under a Curve . . . . . . . . . . . . . . . 904

C H A P T E R 14 Limits and Continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90814.1 Limits of Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90914.2 Properties of Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 918

14.2.A Excursion: One-Sided Limits . . . . . . . . . . . . . . 92414.3 The Formal Definition of Limit (Optional) . . . . . . . . . . . . . . 92914.4 Continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93614.5 Limits Involving Infinity . . . . . . . . . . . . . . . . . . . . . . . . . . . 948

Chapter Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 960can do calculus Riemann Sums . . . . . . . . . . . . . . . . . . . 964

Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 968

Algebra Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 969Advanced Topics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 994Geometry Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1011Technology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1018Glossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1035Answers to Selected Exercises . . . . . . . . . . . . . . . . . . . . 1054Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1148

XVII

APPLICATIONS

Profits, 30, 42, 146, 170, 220, 222,233, 237

Revenue functions, 125, 146, 171,824

Sales, 20, 41, 45, 126, 171-2, 221,344, 363, 404, 790, 803, 812, 833,837

Telemarketing, 848Textbook publication, 42

Chemistry, Physics,and GeologyAntifreeze, 104-5Atmospheric pressure, 388Boiling points, 41, 335Bouncing balls, 16, 62, 64Carbon dioxide, 387Evaporation, 195Fahrenheit v. Celsius, 214Floating balloons, 70Gas pressure, 148Gravitational acceleration, 293Gravity, 677, 680, 687Halleys Comet, 700, 754Light, 117, 602Mixtures, 104-5, 790, 835, 837Orbits of astronomical objects,

393, 697, 700Ozone, 387Pendulums, 335, 556, 602Photography, 293, 538Purification, 350, 355Radio waves, 500Radioactive decay, 340, 352, 355-6,

381, 387, 404, 407Richter Scale, 370, 407Rotating wheels, 442-3, 550, 556Sinkholes, 623Sledding, 678, 687Springs, 551-2, 564Swimming pool chlorine levels, 20Vacuum pumps, 64

Biology and LifeScienceAnimal populations, 21, 223, 233,

342, 344, 355, 388, 407Bacteria, 198, 259, 344, 350, 355,

388Birth rates, 344, 397Blood flow, 221Illness, 42, 55, 88, 273, 362Food, 844, 897Food web, 810, 813Forest fires, 635Genealogy, 64Infant mortality rates, 398Life expectancy, 71, 344Medicine, 223, 292Murder rates, 258, 278Plant growth, 355, 873, 889Population growth, 340, 349, 355,

382-3, 388, 391, 395, 397-8, 404,407, 565

Business andManufacturingArt galleries, 883Cash flows, 44Clothing design, 883Computer speed, 864Cost functions, 123, 125, 149, 181,

183, 292, 319, 325, 831Depreciation, 35, 42, 64, 72Farming, 51, 399-400Food, 49, 54, 105, 790, 803, 813,

833, 851, 883, 897Furniture, 803, 808Gift boxes, 820Gross Domestic Product (GDP),

53Managerial jobs, 73Manufacturing, 136, 216, 221Merchandise production, 20, 42,

126, 135, 400, 789, 831

Weather, 105, 116, 407, 485,553, 557-8, 897, 903

Weather balloons, 198Weight, 116Whispering Gallery, 696

ConstructionAntenna towers, 620Building a ramp, 40Bus shelters, 635Equipment, 30, 106, 428, 431,

442, 469, 636Fencing, 149, 292, 325, 635Highway spotlights, 538Lifting beams, 442Monument construction, 30Paving, 102, 105, 473Pool heaters, 789Rain gutters, 468Sub-plots, 126Surveying, 284Tunnels, 623

Consumer AffairsCampaign contributions, 5Cars, 20, 126Catalogs, 879College, 10, 75, 87, 149, 320,

813Credit cards, 64, 353, 800Energy use, 57, 125Expenditures per student, 355Gas prices, 11Health care, 42Inflation, 355Lawsuits, 864Libraries, 883Life insurance premiums, 53,

72Loans, 56, 72, 105, 353, 803Lotteries, 881, 886

XVIII Applications

Applications XIX

National debt, 41, 399Nuts, 796Poverty levels, 54Property crimes, 276Real estate, 854-5Running Shoes, 898Salaries, 17, 20, 28, 30, 55, 64, 74-5,

105, 126, 388, 403, 864Shipping, 12, 230, 813Stocks, 20, 275, 339Take-out food, 56Taxes, 12, 149-50Television viewing, 344Tickets, 42, 228, 786, 789, 803, 835Unemployment rates, 70, 557

FinancialAppreciation, 72Charitable donations, 74Compound interest, 69, 345-8,

353-4, 360, 362, 382, 387, 404,407

Income, 54-5, 277Investments, 100, 105, 126, 135,

789-90, 803, 833Land values, 789Personal debt, 278Savings, 11, 354

GeometryBeacons, 485Boxes, 149, 161, 259, 292, 323, 325,

825Canal width, 428Cylinders, 149, 161, 292, 325Distance from a falling object, 198Flagpoles, 427, 433, 684Flashlights, 715Footbridges, 700Gates, 622Hands of a clock, 439, 443, 467

Heights of objects, 432, 470-1, 640Leaning Tower of Pisa, 635Ohio Turnpike, 431Paint, 637, 803Parks, 699Rectangles, 149, 161, 169, 171, 825Rectangular rooms, 825Satellite dishes, 715Shadow length, 198, 635, 684Sports, 622, 727, 759-60, 764Tunnels, 471Vacant lots, 637Very Large Array, 713Wagons, 680Water balloons, 216, 221Wires, 427, 431, 825

MiscellaneousAuditorium seating, 20, 30Auto racing, 70Baseball, 864, 883Basketball, 885Birthdays, 883-4Bread, 107Circus Animals, 668Committee officers, 883Darts, 875, 889Drawbridge, 432Dreidels, 882-3Drinking glasses, 432DWI, 388Education, 20, 277, 362, 398-9, 849,

851, 864, 883, 887-8, 894, 897EMS response time, 895FDA drug approval, 56Gears, 442Ice cream preferences, 844-6Latitudes, 442License plates, 883Money, 802-3, 835Oil spills, 67, 231

Psychic abilities, 883Rope, 117, 624, 665, 668Software learning curves, 404Swimming pools, 431Tightrope walking, 116Typing speeds, 388Violent crime rates, 277

Time and DistanceAccident investigation, 335,

859Airplanes, 101, 431-2, 622-4,

629, 636, 666, 668-9, 685-6Altitude and elevation, 427,

431-3, 465, 537, 546, 631,636

Balls, 547, 764Bicycles, 443Boats, 113, 432, 465, 623, 684,

724, 727Buoys, 432Distances, 623, 635-6Explosions, 705, 727Free-falling bodies, 958Helicopters, 117Hours of daylight, 546Lawn mowing, 680Lightning, 708Merry-go-rounds, 440, 442,

556Pedestrian bridges, 433Projectiles, 172, 547, 565, 602,

764Rockets, 106, 126, 172Rocks, 215, 219, 233Satellites, 624Swimming, 669Trains, 619, 622, 684Trucks / cars, 105-6, 135,

221-2, 433, 964-6Walking, 149

Applications XIX

2

On a Clear Day

Hot-air balloons rise linearly as they ascend to the designated height. The distance theyhave traveled, as measured along the ground, is a function of time and can be found byusing a linear function. See Exercise 58 in Chapter 1 Review.

Number Patterns

C H A P T E R

1

3

1.1 Real Numbers, Relations, and Functions

1.2 Mathematical Patterns

1.3 Arithmetic Sequences

1.4 Lines

1.5 Linear Models

1.6 Geometric Sequences

Chapter Review

can do calculus Infinite Geometric Series

Chapter OutlineInterdependence of Sections1.1 1.2 1.3 1.4 1.5

1.6

Mathematics is the study of quantity, order, and relationships. Thischapter defines the real numbers and the coordinate plane, and ituses the vocabulary of relations and functions to begin the study of math-

ematical relationships. The number patterns in recursive, arithmetic, and

geometric sequences are examined numerically, graphically, and algebraic-

ally. Lines and linear models are reviewed.

1.1 Real Numbers, Relations, and Functions

Real number relationships, the points of a line, and the points of a planeare powerful tools in mathematics.

Real Numbers

You have been using real numbers most of your life. Some subsets of thereal numbers are the natural numbers, and the whole num-bers, which include 0 together with the set of natural numbers. Theintegers are the whole numbers and their opposites.

The natural numbers are also referred to as the counting numbers andas the set of positive integers, and the whole numbers are also referredto as the set of nonnegative integers.

p , 5, 4, 3, 2, 1, 0, 1, 2, 3, 4, 5, p

1, 2, 3, 4, p ,

Objectives

Define key terms:sets of numbersthe coordinate planerelationinput and outputdomain and rangefunction

Use functional notation

> > > >>

A real number is said to be a rational number if it can be expressed as a

ratio, with r and s integers and The following are rational numbers.

and

Rational numbers may also be described as numbers that can be expressed

as terminating decimals, such as or as nonterminating repeat-

ing decimals in which a single digit or block of digits repeats, such as

or

A real number that cannot be expressed as a ratio with integer numera-tor and denominator is called an irrational number. Alternatively, anirrational number is one that can be expressed as a nonterminating, non-repeating decimal in which no single digit or block of digits repeats. Forexample, the number which is used to calculate the area of a circle, isirrational.

Although Figure 1.1-1 does not represent the size of each set of numbers,it shows the relationship between subsets of real numbers.

All natural numbers are whole numbers. All whole numbers are integers. All integers are rational numbers. All rational numbers are real numbers. All irrational numbers are real numbers.

p,

53333 0.159159 p

53 1.6666 p

0.25 14 ,

8 35 43547

471 ,0.983

9381000,

12,

s 0.rs ,

4 Chapter 1 Number Patterns

Figure 1.1-1

Rational Numbers

Real Numbers

Irrational Numbers

Integers

Whole Numbers

Natural Numbers

Figure 1.1-2

2345678

8.6

9 654 98732101

5.78 2.2 726.33

2337

The Real Number Line

The real numbers are represented graphically as points on a number line,as shown in Figure 1.1-2. There is a one-to-one correspondence betweenthe real numbers and the points of the line, which means that each realnumber corresponds to exactly one point on the line, and vice versa.

The Coordinate Plane

Just as real numbers correspond to points on a number line, ordered pairsof real numbers correspond to points in a plane. To sketch a coordinateplane, draw two number lines in the plane, one vertical and one hori-zontal, as shown in Figure 1.1-3.

The number lines, or axes, are often named the x-axis and the y-axis, butother letters may be used. The point where the axes intersect is the origin,and the axes divide the plane into four regions, called quadrants,indicated by Roman numerals in Figure 1.1-3. The plane is now said tohave a rectangular, or Cartesian, coordinate system.

In Figure 1.1-3, point P is represented by an ordered pair that hascoordinates (c, d), where c is the x-coordinate of P, and d is the y-coordinate of P.

Scatter Plots

In many application problems, data is plotted as points on the coordinateplane. This type of representation of data is called a scatter plot.

Example 1 Scatter Plot

Create a scatter plot of this data from the Federal Election Commissionthat shows the total amount of money, in millions of dollars, contributedto all congressional candidates in the years shown.

Section 1.1 Real Numbers, Relations, and Functions 5

Figure 1.1-3

II I

III IV

P

c x

y

d

Year 1988 1990 1992 1994 1996

Amount 276 284 392 418 500

Solution

Let x be the number of years since 1988, so that denotes 1988, denotes 1990, and so on. Plot the points (0, 276), (2, 284), (4, 392), (6, 418),and (8, 500) to obtain a scatter plot. See Figure 1.1-4.

x 2x 0

Figure 1.1-4

500

y

x

400

300

200

104 862

100

0

Years since 1988

Am

ount

s(i

n m

illio

ns o

f dol

lars

)

A Relation and Its Domain and Range

Scientists and social scientists spend much time and money looking forhow two quantities are related. These quantities might be a personsheight and his shoe size or how much a person earns and how many yearsof college she has completed. In these examples, a relation exists betweentwo variables. The first quantity, often called the x-variable, is said to berelated to the second quantity, often called the y-variable. Mathematiciansare interested in the types of relations that exist between two quantities,or how x and y are paired. Of interest is a relations domain, or possiblevalues that x can have, as well as a relations range, possible values thaty can have. Relations may be represented numerically by a set of orderedpairs, graphically by a scatter plot, or algebraically by an equation.

Example 2 Domain and Range of a Relation

The table below shows the heights and shoe sizes of twelve high schoolseniors.

6 Chapter 1 Number Patterns

Height67 72 69 76 67 72

(inches)

Shoe size 8.5 10 12 12 10 11

Height67 62.5 64.5 64 62 62

(inches)

Shoe Size 7.5 5.5 8 8.5 6.5 6

For convenience, the data table lists the height first, so the pairing (height,shoe size) is said to be ordered. Hence, the data is a relation. Find therelations domain and range.

Solution

There are twelve ordered pairs.

Figure 1.1-5 shows the scatter plot of the relation.

The relations domain is the set of x values: {62, 62.5, 64, 64.5, 67, 69, 72,76}, and its range is the set of y values: {5.5, 6, 6.5, 7.5, 8, 8.5, 10, 11, 12}.

167, 7.52, 162.5, 5.52, 164.5, 82, 164, 8.52, 162, 6.52, 162, 62167, 8.52, 172, 102, 169, 122, 176, 122, 167, 102, 172, 112,

Figure 1.1-5

12

Shoe

Siz

e

Height

10

8

6

72 7624 6036 4812

4

2

Sometimes a rule, which is a statement or an equation, expresses onequantity in a relation in terms of the other quantity.

Example 3 A Rule of a Relation

Given the relation

state its domain and range. Create a scatter plot of the relation and finda rule that relates the value of the first coordinate to the value of the sec-ond coordinate.

Solution

The domain is {0, 1, 4, 9}, and the range is Thescatter plot is shown in Figure 1.1-6. One rule that relates the first coor-dinate to the second coordinate in each pair is where x is an integer.

Functions

Much of mathematics focuses on special relations called functions. Afunction is a set of ordered pairs in which the first coordinate denotes theinput, the second coordinate denotes the output that is obtained from therule of the function, and

each input corresponds to one and only one output.

Think of a function as a calculator with only one key that provides thesolution for the rule of the function. A number is input into the calculator,the rule key (which represents a set of operations) is pushed, and a sin-gle answer is output to the display. On the special function calculator,shown in Figure 1.1-7, if you press 9 then the display screen willshow 163twice the square of 9 plus 1. The number 9 is the input, therule is given by and the output is 163.

Example 4 Identifying a Function Represented Numerically

In each set of ordered pairs, the first coordinate represents input and thesecond coordinate represents its corresponding output. Explain why eachset is, or is not, a function.

a.

b.

c.

Solution

The phrase one and only one in the definition of a function is the crit-ical qualifier. To determine if a relation is a function, make sure that eachinput corresponds to exactly one output.

5 10, 02, 11, 12, 11, 12, 14, 22, 14, 22, 19, 32, 19, 32 65 10, 02, 11, 12, 11, 12, 14, 22, 14, 22, 19, 32, 19, 32 65 10, 02, 11, 12, 11, 12, 14, 22, 14, 22, 19, 32, 19, 32 6

2x2 1,

2x2 1,

x y2,

53, 2, 1, 0, 1, 2, 36.

5 10, 02, 11, 12, 11, 12, 14, 22, 14, 22, 19, 32, 19, 32 6,

Section 1.1 Real Numbers, Relations, and Functions 7

Figure 1.1-6

x

y

4

2

4

2

104 862

Figure 1.1-7

a. is not a functionbecause the input 1 has two outputs, 1 and Two other inputs, 4and 9, also have more than one output.

b. is a function.Although 1 appears as an output twice, each input has one and onlyone output.

c. is a functionbecause each input corresponds to one and only one output.

5 10, 02, 11, 12, 11, 12, 14, 22, 14, 22, 19, 32, 19, 32 6

5 10, 02, 11, 12, 11, 12, 14, 22, 14, 22, 19, 32, 19, 32 61.

5 10, 02, 11, 12, 11, 12, 14, 22, 14, 22, 19, 32, 19, 32 6

8 Chapter 1 Number Patterns

Example 5 Finding Function Values from a Graph

The graph in Figure 1.1-8 defines a function whose rule is:

For input x, the output is the unique number y such that (x, y) is on the graph.

Figure 1.1-8

2

2 4

4

242

x

y

Calculator Exploration

Make a scatter plot of each set of ordered pairs in Example 4. Exam-ine each scatter plot to determine if there is a graphical test that canbe used to determine if each input produces one and only one out-put, that is, if the set represents a function.

a. Find the output for the input 4.b. Find the inputs whose output is 0.c. Find the y-value that corresponds to d. State the domain and range of the function.

Solution

a. From the graph, if then Therefore, 3 is the outputcorresponding to the input 4.

y 3.x 4,

x 2.

b. When or Therefore, and 2 arethe inputs corresponding to the output 0.

c. The y-value that corresponds to is d. The domain is all real numbers between and 5, inclusive. The

range is all real numbers between and 3, inclusive.

Function NotationBecause functions are used throughout mathematics, function notationis a convenient shorthand developed to make their use and analysis eas-ier. Function notation is easily adapted to mathematical settings, in whichthe particulars of a relationship are not mentioned. Suppose a function isgiven. Let f denote a given function and let a denote a number in thedomain of f. Then

denotes the output of the function f produced by input a.

For example, f(6) denotes the output of the function f that corresponds tothe input 6.

y is the output produced by input x according to the rule of the function f

is abbreviated

which is read y equals f of x.

In actual practice, functions are seldom presented in the style of domain-rule-range, as they have been here. Usually, a phrase, such as thefunction will be given. It should be understood as a setof directions, as shown in the following diagram.

f 1x2 2x2 1,

y f 1x2,

f(a)

24

y 3.x 2

3, 0,x 0 or x 2.y 0, x 3

Section 1.1 Real Numbers, Relations, and Functions 9

CAUTION

The parentheses in d(t)do not denote multi-plication. The entiresymbol d(t) is part ofthe shorthand languagethat is convenient forrepresenting a function,its input and its output;it is not the same asalgebraic notation.

The choice ofletters that represent thefunction and input mayvary.

NOTE

Name of function Input number

Output number Directions that tell you what to do with input

x in order to produce the corresponding output

namely, square it, add 1, and take the

square root of the result.

f(x),

f 1x2 2x2 1>

>

>

>

For example, to find f(3), the output of the function f for input 3, simplyreplace x by 3 in the rules directions.

Similarly, replacing x by and 0 produces the respective outputs.

f 152 21522 1 226 and f 102 202 1 15

f132 21322 1 29 1 210 f 1x2 2x2 1

Example 6 Function Notation

For find each of the following:

a. b. c.

Solution

To find and replace x by and respectively, in therule of h.

a.

b.

The values of the function h at any quantity, such as can be found byusing the same procedure: replace x in the formula for h(x) by the quantity

and simplify.

c.

h1a2 1a22 1a2 2 a2 a 2a

a,

h122 1222 122 2 4 2 2 0h A23 B A23 B 2 23 2 3 23 2 1 23

2,23h 122,h A23 B

h1a2h122h A23 Bh1x2 x2 x 2,

10 Chapter 1 Number Patterns

Exercises 1.1

1. Find the coordinates of points AI. In Exercises 68, sketch a scatter plot of the given data.In each case, let the x-axis run from 0 to 10.

6. The maximum yearly contribution to anindividual retirement account in 2003 is $3000.The table shows the maximum contribution infixed 2003 dollars. Let correspond to 2000.x 0

I

C

D

E

BA

G

H1

2

3

321

F

y

x

In Exercises 25, find the coordinates of the point P.

2. P lies 4 units to the left of the y-axis and 5 unitsbelow the x-axis.

3. P lies 3 units above the x-axis and on the samevertical line as

4. P lies 2 units below the x-axis and its x-coordinateis three times its y-coordinate.

5. P lies 4 units to the right of the y-axis and its y-coordinate is half its x-coordinate.

16, 72.

Year 2003 2004 2005 2006 2007 2008

Amount 3000 2910 3764 3651 3541 4294

7. The table shows projected sales, in thousands, ofpersonal digital video recorders. Let correspond to 2000. (Source: eBrain MarketResearch)

x 0

8. The tuition and fees at public four-year colleges inthe fall of each year are shown in the table. Let

correspond to 1995. (Source: The CollegeBoard)x 0

Year 2000 2001 2002 2003 2004 2005

Sales 257 129 143 214 315 485

Technology Tip

One way to evaluate afunction is to enter

its rule into the equationmemory as anduse TABLE or EVAL. Seethe Technology Appendixfor more detailed information.

y f 1x2f 1x2

Functions will bediscussed in detail inChapter 3.

NOTE

9. The graph, which is based on data from the U.S.Department of Energy, shows approximate averagegasoline prices (in cents per gallon) between 1985and 1996, with corresponding to 1985.x 0

Section 1.1 Real Numbers, Relations, and Functions 11

a. In what years during this period were personalsavings largest and smallest (as a percent ofdisposable income)?

b. In what years were personal savings at least7% of disposable income?

11. a. If the first coordinate of a point is greater than3 and its second coordinate is negative, in whatquadrant does it lie?

b. What is the answer to part a if the firstcoordinate is less than 3?

12. In which possible quadrants does a point lie if theproduct of its coordinates is a. positive? b. negative?

13. a. Plot the points (3, 2), and

b. Change the sign of the y-coordinate in each ofthe points in part a, and plot these new points.

c. Explain how the points (a, b) and arerelated graphically.Hint: What are their relative positions withrespect to the x-axis?

14. a. Plot the points (5, 3), and

b. Change the sign of the x-coordinate in each ofthe points in part a, and plot these new points.

c. Explain how the points (a, b) and arerelated graphically. Hint: What are their relative positions withrespect to the y-axis?

In Exercises 15 18, determine whether or not the giventable could possibly be a table of values of a function.Give reasons for each answer.

15.

1a, b2

13, 52.11, 42,14, 22,

1a, b2

15, 42.12, 32,14, 12,

40

6 7 8 9 10 111 2 3 4 5

20

60

100

80

120

x

y

54

30 355 10 15 20 25

321

6

10987

x

y

Input 2 0 3 1 5

Output 2 3 2.5 2 14

Input 5 3 0 3 5

Output 7 3 0 5 3

Input 5 1 3 5 7

Output 0 2 4 6 8

Input 1 1 2 2 3

Output 1 2 5 6 8

16.

17.

18.

a. Estimate the average price in 1987 and in 1995.b. What was the approximate percentage increase

in the average price from 1987 to 1995?c. In what year(s) was the average price at least

$1.10 per gallon?

10. The graph, which is based on data from the U.S.Department of Commerce, shows the approximateamount of personal savings as a percent ofdisposable income between 1960 and 1995, with

corresponding to 1960.x 0

TuitionYear & fees

1995 $2860

1996 $2966

1997 $3111

TuitionYear & fees

1998 $3247

1999 $3356

2000 $3510

19. Find the output (tax amount) that is produced byeach of the following inputs (incomes):$500 $1509 $3754$6783 $12,500 $55,342

20. Find four different numbers in the domain of thisfunction that produce the same output (number inthe range).

21. Explain why your answer in Exercise 20 does notcontradict the definition of a function.

22. Is it possible to do Exercise 20 if all four numbersin the domain are required to be greater than2000? Why or why not?

23. The amount of postage required to mail a first-class letter is determined by its weight. In thissituation, is weight a function of postage? Or viceversa? Or both?

24. Could the following statement ever be the rule ofa function?

For input x, the output is the number whose square is x.

Why or why not? If there is a function with thisrule, what is its domain and range?

Use the figure at the top of page 13 for Exercises 2531. Each of the graphs in the figure defines a func-tion.

25. State the domain and range of the functiondefined by graph a.

26. State the output (number in the range) that thefunction of Exercise 25 produces from thefollowing inputs (numbers in the domain):2, 1, 0, 1.

12 Chapter 1 Number Patterns

27. Do Exercise 26 for these numbers in the

domain:

28. State the domain and range of the functiondefined by graph b.

29. State the output (number in the range) that thefunction of Exercise 28 produces from thefollowing inputs (numbers in the domain):

30. State the domain and range of the functiondefined by graph c.

31. State the output (number in the range) that thefunction of Exercise 30 produces from thefollowing inputs (numbers in the domain):

32. Find the indicated values of the function by handand by using the table feature of a calculator.

a. b. g(0) c. g(4)d. g(5) e. g(12)

33. The rule of the function f is given by the graph.Finda. the domain of fb. the range of fc.d.e.f.

34. The rule of the function g is given by the graph.Finda. the domain of gb. the range of gc.d.e.f. g 142

g 112g 112g 132

f 122f 112f 112f 132

g 122g 1x2 2x 4 2

2, 1, 0, 12, 1.

2, 0, 1, 2.5, 1.5.

12,

52,

52.

1

2

3

234

2

3

3 421

4

x

y

1

1

2

3

234

2

3

3 421

4

x

y

1

Exercises 1922 refer to the following state income taxtable.

Annual income Amount of tax

Less than $2000 0

$2000$6000 2% of income over $2000

More than $6000 $80 plus 5% of incomeover $6000

Section 1.2 Mathematical Patterns 13

1

1

2

34

23

2

3

4

32

a.

1

x

y

1

1

2

34

23

2

3

4

3 42

b.

1

x

y

1

1

2

34

21 1 13

2

3

4

32

c.

1

x

y

An infinite sequence is a sequence with an infinite number of terms.Examples of infinite sequences are shown below.

The three dots, or points of ellipsis, at the end of a sequence indicate thatthe same pattern continues for an infinite number of terms.

A special notation is used to represent a sequence.

2, 1, 23, 24,

25,

26,

27, p

1, 3, 5, 7, 9, 11, 13, p2, 4, 6, 8, 10, 12, p

A sequence is an ordered list of numbers.Each number in the list is called a term of the sequence.

Definition of aSequence

1.2 Mathematical Patterns

Visual patterns exist all around us, and many inventions and discoveriesbegan as ideas sparked by noticing patterns.

Consider the following lists of numbers.

Analyzing the lists above, many people would say that the next numberin the list on the left is 11 because the pattern appears to be add 3 to theprevious term. In the list on the right, the next number is uncertainbecause there is no obvious pattern. Sequences may help in the visuali-zation and understanding of patterns.

1, 10, 3, 73, ?4, 1, 2, 5, 8, ?

Objectives

Define key terms:sequencesequence notationrecursive functions

Create a graph of asequence

Apply sequences to real-world situations

Example 1 Terms of a Sequence

Make observations about the pattern suggested by the diagrams below.Continue the pattern by drawing the next two diagrams, and write asequence that represents the number of circles in each diagram.

Diagram 1 Diagram 2 Diagram 31 circle 3 circles 5 circles

Solution

Adding two additional circles to the previous diagram forms each newdiagram. If the pattern continues, then the number of circles in Diagram4 will be two more than the number of circles in Diagram 3, and thenumber of circles in Diagram 5 will be two more than the number inDiagram 4.

Diagram 4 Diagram 57 circles 9 circles

The number of circles in the diagrams is represented by the sequence

which can be expressed using sequence notation.

u5 9 p un un1 2u4 7u3 5u2 3u1 1

51, 3, 5, 7, 9, p 6,

14 Chapter 1 Number Patterns

The following notation denotes specific terms of a sequence:

The first term of a sequence is denoted

The second term The term in the nth position, called the nth term, is

denoted by

The term before is un1.un

un.

u2.u1.

SequenceNotation

Any letter can beused to represent the termsof a sequence.

NOTE

Technology Tip

If needed, review howto create a scatter plot

in the Technology Appendix.

Graphs of Sequences

A sequence is a function, because each input corresponds to exactly oneoutput.

The domain of a sequence is a subset of the integers. The range is the set of terms of the sequence.

Because the domain of a sequence is discrete, the graph of a sequenceconsists of points and is a scatter plot.

Example 2 Graph of a Sequence

Graph the first five terms of the sequence .

Solution

The sequence can be written as a set of ordered pairs where the first coor-dinate is the position of the term in the sequence and the secondcoordinate is the term.

(1, 1) (2, 3) (3, 5) (4, 7) (5, 9)

The graph of the sequence is shown in Figure 1.2-1.

Recursive Form of a Sequence

In addition to being represented by a listing or a graph, a sequence canbe denoted in recursive form.

51, 3, 5, 7, 9, p 6

Section 1.2 Mathematical Patterns 15

Figure 1.2-10

10

0 10

Figure 1.2-210

10

0 10

Example 3 Recursively Defined Sequence

Define the sequence recursively and graph it.

Solution

The sequence can be expressed as

The first term is given. The second term is obtained by adding 3 to thefirst term, and the third term is obtained by adding 3 to the second term.Therefore, the recursive form of the sequence is

and for

The ordered pairs that denote the sequence are

The graph is shown in Figure 1.2-2.

11, 72 12, 42 13, 12 14, 22 15, 52

n 2.un un1 3u1 7

u1 7 u2 4 u3 1 u4 2 u5 5

57, 4, 1, 2, 5, p 6

A sequence is defined recursively if the first term is givenand there is a method of determining the nth term by usingthe terms that precede it.

RecursivelyDefined Sequence

Alternate Sequence NotationSometimes it is more convenient to begin numbering the terms of asequence with a number other than 1, such as 0 or 4.

or

Example 4 Using Alternate Sequence Notation

A ball is dropped from a height of 9 feet. It hits the ground and bouncesto a height of 6 feet. It continues to bounce, and on each rebound it rises

to the height of the previous bounce.

a. Write a recursive formula for the sequence that represents the heightof the ball on each bounce.

b. Create a table and a graph showing the height of the ball on eachbounce.

c. Find the height of the ball on the fourth bounce.

Solution

a. The initial height, is 9 feet. On the first bounce, the rebound

height, is 6 feet, which is the initial height of 9 feet. The

recursive form of the sequence is given by

b. Set the mode of the calculator to Seq instead of Func and enter thefunction as shown on the next page in Figure 1.2-4a. Figure 1.2-4bdisplays the table of values of the function, and Figure 1.2-4cdisplays the graph of the function.

u0 9 and un 23 un1 for n 1

23u1,

u0,

23

b4, b5, b6, pu0, u1, u2, p

16 Chapter 1 Number Patterns

Figure 1.2-3

Technology Tip

The sequence graphingmode can be found in

the TI MODE menu or theRECUR submenu of theCasio main menu. On suchcalculators, recursivelydefined function may beentered into the sequencememory, or Checkyour instruction manualfor the correct syntax and use.

Y list.

Calculator Exploration

An alternative way to think about the sequence in Example 3 is

Each

Type into your calculator and press ENTER. Thisestablishes the first answer.

To calculate the second answer, press to automatically placeANS at the beginning of the next line of the display.

Now press 3 and ENTER to display the second answer. Pressing ENTER repeatedly will display subsequent answers.

See Figure 1.2-3.

7

answer Preceding answer 3.

c. As shown in Figures 1.2-4b and 1.2-4c, the height on the fourthbounce is approximately 1.7778 feet.

Applications using Sequences

Example 5 Salary Raise Sequence

If the starting salary for a job is $20,000 and a raise of $2000 is earned atthe end of each year of work, what will the salary be at the end of thesixth year? Find a recursive function to represent this problem and use atable and a graph to find the solution.

Solution

The initial term, is 20,000. The amount of money earned at the end ofthe first year, , will be 2000 more than The recursive function

will generate the sequence that represents the salaries for each year. Asshown in Figures 1.2-5a and 1.2-5b, the salary at the end of the sixth yearwill be $32,000.

u0 20,000 and un un1 2000 for n 1

u0.u1u0,

Section 1.2 Mathematical Patterns 17

Figure 1.2-4a Figure 1.2-4b Figure 1.2-4c

10

0

0 10

Figure 1.2-5a Figure 1.2-5b

50,000

0

0 10

In the previous examples, the recursive formulas were obtained by eitheradding a constant value to the previous term or by multiplying the pre-vious term by a constant value. Recursive functions can also be obtainedby adding different values that form a pattern.

Example 6 Sequence Formed by Adding a Pattern of Values

A chord is a line segment joining two points of a circle. The following dia-gram illustrates the maximum number of regions that can be formed by1, 2, 3, and 4 chords, where the regions are not required to have equalareas.

18 Chapter 1 Number Patterns

1 Chord

2 Regions 4 Regions 7 Regions 11 Regions

2 Chords 3 Chords 4 Chords

a. Find a recursive function to represent the maximum number ofregions formed with n chords.

b. Use a table to find the maximum number of regions formed with 20chords.

Solution

Let the initial number of regions occur with 1 chord, so . The max-imum number of regions formed for each number or chords is shown inthe following table.

Number of chords Maximum number of regions1234

The recursive function is shown as the last entry in the listing above, andthe table and graph, as shown in Figures 1.2-6a and 1.2-6b, identify the20th term of the sequence as 211. Therefore, the maximum number ofregions that can be formed with 20 chords is 211.

Example 7 Adding Chlorine to a Pool

Dr. Miller starts with 3.4 gallons of chlorine in his pool. Each day he adds0.25 gallons of chlorine and 15% evaporates. How much chlorine will bein his pool at the end of the sixth day?

Solution

The initial amount of chlorine is 3.4 gallons, so and each day 0.25gallons of chlorine are added. Because 15% evaporates, 85% of the mix-ture remains.

u0 3.4

un un1 nnpp

u4 11 u3 4u3 7 u2 3u2 4 u1 2u1 2

u1 2

Figure 1.2-6b

Figure 1.2-6a

300

0

0 21

The amount of chlorine in the pool at the end of the first day is obtainedby adding 0.25 to 3.4 and then multiplying the result by 0.85.

The procedure is repeated to yield the amount of chlorine in the pool atthe end of the second day.

Continuing with the same pattern, the recursive form for the sequence is

and for

As shown in Figures 1.2-7a and 1.2-7b, approximately 2.165 gallons ofchlorine will be in the pool at the end of the sixth day.

n 1.un 0.851un1 0.252u0 3.4

0.8513.1025 0.252 2.85

0.8513.4 0.252 3.1025

Section 1.2 Mathematical Patterns 19

Figure 1.2-7b

Figure 1.2-7a

4

0

0 10

Exercises 1.2

In Exercises 14, graph the first four terms of thesequence.

1.

2.

3.

4.

In Exercises 58, define the sequence recursively andgraph the sequence.

5.

6.

7.

8.

In Exercises 912, find the first five terms of the givensequence.

9. and for

10. and for n 2un 13 un1 4u1 5

n 2un 2un1 3u1 4

e8, 4, 2, 1, 12, 14, p f56, 11, 16, 21, 26, p 654, 8, 16, 32, 64, p 656, 4, 2, 0, 2, p 6

54, 12, 36, 108, p 654, 5, 8, 13, p .653, 6, 12, 24, p .652, 5, 8, 11, p .6

11. andfor

12. and for

13. A really big rubber ball will rebound 80% of itsheight from which it is dropped. If the ball isdropped from 400 centimeters, how high will itbounce after the sixth bounce?

14. A tree in the Amazon rain forest grows an averageof 2.3 cm per week. Write a sequence thatrepresents the weekly height of the tree over thecourse of 1 year if it is 7 meters tall today. Write arecursive formula for the sequence and graph thesequence.

15. If two rays have a common endpoint, one angle isformed. If a third ray is added, three angles areformed. See the figure below.

n 2un nun1u0 1, u1 1

n 4un un1 un2 un3u1 1, u2 2, u3 3,

12 3

Write a recursive formula for the number ofangles formed with n rays if the same patterncontinues. Graph the sequence. Use the formula tofind the number of angles formed by 25 rays.

16. Swimming pool manufacturers recommend thatthe concentration of chlorine be kept between 1and 2 parts per million (ppm). They also warnthat if the concentration exceeds 3 ppm,swimmers experience burning eyes. If theconcentration drops below 1 ppm, the water willbecome cloudy. If it drops below 0.5 ppm, algaewill begin to grow. During a period of one day15% of the chlorine present in the pool dissipates,mainly due to evaporation.a. If the chlorine content is currently 2.5 ppm and

no additional chlorine is added, how long willit be before the water becomes cloudy?

b. If the chlorine content is currently 2.5 ppm and0.5 ppm of chlorine is added daily, what willthe concentration eventually become?

c. If the chlorine content is currently 2.5 ppm and0.1 ppm of chlorine is added daily, what willthe concentration eventually become?

d. How much chlorine must be added daily forthe chlorine level to stabilize at 1.8 ppm?

17. An auditorium has 12 seats in the front row. Eachsuccessive row, moving towards the back of theauditorium, has 2 additional seats. The last rowhas 80 seats.

Write a recursive formula for the number of seatsin the nth row and use the formula to find thenumber of seats in the 30th row.

18. In 1991, the annual dividends per share of a stockwere approximately $17.50. The dividends wereincreasing by $5.50 each year. What were theapproximate dividends per share in 1993, 1995,and 1998? Write a recursive formula to representthis sequence.

19. A computer company offers you a job with astarting salary of $30,000 and promises a 6% raiseeach year. Find a recursive formula to representthe sequence, and find your salary ten years fromnow. Graph the sequence.

20. Book sales in the United States (in billions ofdollars) were approximated at 15.2 in the year1990. The book sales increased by 0.6 billion eachyear. Find a sequence to represent the book salesfor the next four years, and write a recursiveformula to represent the sequence. Graph thesequence and predict the number of book sales in2003.

21. The enrollment at Tennessee State University iscurrently 35,000. Each year, the school willgraduate 25% of its students and will enroll 6,500

20 Chapter 1 Number Patterns

new students. What will be the enrollment 8 yearsfrom now?

22. Suppose you want to buy a new car and finance itby borrowing $7,000. The 12-month loan has anannual interest rate of 13.25%.a. Write a recursive formula that provides the

declining balances of the loan for a monthlypayment of $200.

b. Write out the first five terms of this sequence.c. What is the unpaid balance after 12 months?d. Make the necessary adjustments to the monthly

payment so that the loan can be paid off in 12equal payments. What monthly payment isneeded?

23. Suppose a flower nursery manages 50,000 flowersand each year sells 10% of the flowers and plants4,000 new ones. Determine the number of flowersafter 20 years and 35 years.

24. Find the first ten terms of a sequence whose firsttwo terms are and and whose nthterm is the sum of the two precedingterms.

Exercises 2529 deal with prime numbers. A positiveinteger greater than 1 is prime if its only positive inte-ger factors are itself and 1. For example, 7 is primebecause its only factors are 7 and 1, but 15 is not primebecause it has factors other than 15 and 1, namely, 3and 5.

25. Critical Thinking a. Let be the sequence ofprime integers in their usual ordering. Verifythat the first ten terms are 2, 3, 5, 7, 11, 13, 17,19, 23, 29.b. Find

In Exercises 2629, find the first five terms of thesequence.

26. Critical Thinking is the nth prime integer largerthan 10.

27. Critical Thinking is the square of the nth primeinteger.

28. Critical Thinking is the number of primeintegers less than n.

29. Critical Thinking is the largest prime integer lessthan 5n.

Exercises 3034 deal with the Fibonacci sequence { }which is defined as follows:

un

un

un

un

un

u17, u18, u19, u20.

5un6

1for n 32u2 1u1 1

and for is the sum of the twopreceding terms, That is,

30. Critical Thinking Leonardo Fibonacci discoveredthe sequence in the thirteenth century inconnection with the following problem: A rabbitcolony begins with one pair of adult rabbits, onemale and one female. Each adult pair producesone pair of babies, one male and one female,every month. Each pair of baby rabbits becomesadult and produces its first offspring at age twomonths. Assuming that no rabbits die, how many

u1 1u2 1

u3 u1 uz 1 1 2 u4 u2 u3 1 2 3 u5 u3 u4 2 3 5

and so on.

un un1 un2.n 3, unu1 1, u2 1,

Section 1.3 Arithmetic Sequences 21

adult pairs of rabbits are in the colony at the endof n months, Hint: It may behelpful to make up a chart listing for each monththe number of adult pairs, the number of one-month-old pairs, and the number of baby pairs.

31. Critical Thinking List the first ten terms of theFibonacci sequence.

32. Critical Thinking Verify that every positive integerless than or equal to 15 can be written as aFibonacci number or as a sum of Fibonaccinumbers, with none used more than once.

33. Critical Thinking Verify that is aperfect square for

34. Critical Thinking Verify that for n 2, 3, p , 10.112n1

un1 un1 1un22 n 1, 2, p , 10.

51un22 4112n

n 1, 2, 3, p ?

1.3 Arithmetic Sequences

An arithmetic sequence, which is sometimes called an arithmetic progres-sion, is a sequence in which the difference between each term and thepreceding term is always constant.

Example 1 Arithmetic Sequence

Are the following sequences arithmetic? If so, what is the differencebetween each term and the term preceding it?

a.

b.

Solution

a. The difference between each term and the preceding term is . Sothis is an arithmetic sequence with a difference of .

b. The difference between the 1st and 2nd terms is 2 and the differencebetween the 2nd and 3rd terms is 3. The differences are not constant,therefore this is not an arithmetic sequence.

If is an arithmetic sequence, then for each the term precedingis and the difference is some constantusually called

d. Therefore, un un1 d.un un1un1un

n 2,5un6

44

53, 5, 8, 12, 17, p 6514, 10, 6, 2,2, 6, 10, p 6

Objectives

Identify and graph anarithmetic sequence

Find a common difference

Write an arithmeticsequence recursively andexplicitly

Use summation notation

Find the nth term and thenth partial sum of anarithmetic sequence

The number d is called the common difference of the arithmetic sequence.

Example 2 Graph of an Arithmetic Sequence

If is an arithmetic sequence with and as its firsttwo terms,

a. find the common difference.b. write the sequence as a recursive function.c. give the first seven terms of the sequence.d. graph the sequence.

Solution

a. The sequence is arithmetic and has a common difference of

b. The recursive function that describes the sequence is

and for

c. The first seven terms are 3, 4.5, 6, 7.5, 9, 10.5, and 12, as shown inFigure 1.3-1a.

d. The graph of the sequence is shown in Figure 1.3-1b.

Explicit Form of an Arithmetic Sequence

Example 2 illustrated an arithmetic sequence expressed in recursive formin which a term is found by using preceding terms. Arithmetic sequencescan also be expressed in a form in which a term of the sequence can befound based on its position in the sequence.

Example 3 Explicit Form of an Arithmetic Sequence

Confirm that the sequence with can also beexpressed as

Solution

Use the recursive function to find the first few terms of the sequence.

u2 7 4 3 u1 7

un 7 1n 12 4.u1 7un un1 4

n 2un un1 1.5u1 3

u2 u1 4.5 3 1.5

u2 4.5u1 35un6

22 Chapter 1 Number Patterns

In an arithmetic sequence { }

for some constant d and all n 2.

un un1 d

unRecursive Form

of an ArithmeticSequence

Figure 1.3-1b

15

0

0 10

Figure 1.3-1a

Notice that is which is the first term of the sequence withthe common difference of 4 added twice. Also, is which isthe first term of the sequence with the common difference of 4 addedthree times. Because this pattern continues, Thetable in Figure 1.3-2b confirms the equality of the two functions.

un un1 4 with u1 7 and un 7 1n 12 4un 7 1n 12 4.

7 3 4,u47 2 4,u3

u5 17 3 42 4 7 4 4 7 16 9 u4 17 2 42 4 7 3 4 7 12 5 u3 17 42 4 7 2 4 7 8 1

Section 1.3 Arithmetic Sequences 23

Figure 1.3-3

30

10

0 10

Figure 1.3-2b

Figure 1.3-2a

If the initial term of a sequence is denoted as the explicit form of anarithmetic sequence with common difference d is

Example 4 Explicit Form of an Arithmetic Sequence

Find the nth term of an arithmetic sequence with first term and com-mon difference of 3. Sketch a graph of the sequence.

Solution

Because and the formula in the box states that

The graph of the sequence is shown in Figure 1.3-3.

un u1 1n 12d 5 1n 123 3n 8d 3,u1 5

5

un u0 nd for every n 0.

u0,

In an arithmetic sequence { } with common difference d,

un u1 1n 12d for every n 1.un

Explicit Form ofan Arithmetic

Sequence

As shown in Example 3, if is an arithmetic sequence with commondifference d, then for each can be written as a func-tion in terms of n, the position of the term.

Applying the procedure shown in Example 3 to the general case showsthat

Notice that 4d is added to to obtain . In general, adding toyields So is the sum of n numbers: and the common differ-

ence, d, added times.1n 12 u1unun.u1

1n 12du5u1 u5 u4 d 1u1 3d2 d u1 4d u4 u3 d 1u1 2d2 d u1 3d u3 u2 d 1u1 d2 d u1 2d u2 u1 d

n 2, un un1 d5un6

Example 5 Finding a Term of an Arithmetic Sequence

What is the 45th term of the arithmetic sequence whose first three termsare 5, 9, and 13?

Solution

The first three terms show that and that the common difference,d, is 4. Apply the formula with .

Example 6 Finding Explicit and Recursive Formulas

If is an arithmetic sequence with and find , a recur-sive formula, and an explicit formula for

Solution

The sequence can be written as

The common difference, d, can be found by the difference between 93 and57 divided by the number of times d must be added to 57 to produce 93(i.e., the number of terms from 6 to 10).

Note that is the difference of the output values (terms of thesequence) divided by the difference of the input values (position of theterms of the sequence), which represents the change in output per unitchange in input.

The value of can be found by using and in theequation

Because and the recursive form of the arithmetic sequenceis given by

and for

The explicit form of the arithmetic sequence is given by

9n 3, for n 1.

un 12 1n 129

n 2un un1 9,u1 12

d 9,u1 12

u1 57 5 9 57 45 12 57 u1 16 129 u6 u1 1n 12d

d 9n 6, u6 57u1

d 9

d 93 5710 6 364 9

u6 u7 u8 u9 u10p , 57, , , , 93, p

un.u1u10 93,u6 575un6

u45 u1 145 12d 5 1442 142 181n 45u1 5

24 Chapter 1 Number Patterns

{ { { { {

Summation Notation

It is sometimes necessary to find the sum of various terms in a sequence.For instance, we might want to find the sum of the first nine terms of thesequence Mathematicians often use the capital Greek letter sigma

to abbreviate such a sum as follows.

Similarly, for any positive integer m and numbers ,c1, c2, p , cm

a9

i1ui u1 u2 u3 u4 u5 u6 u7 u8 u9

12 5un6.

Section 1.3 Arithmetic Sequences 25

Example 7 Sum of a Sequence

Compute each sum.

a. b.

Solution

a. Substitute 1, 2, 3, 4, and 5 for n in the expression and addthe terms.

b. Substitute 1, 2, 3, and 4 for n in the expression andadd the terms.

12

3 1 5 9

3 13 42 13 82 13 122 33 0 4 4 33 1 4 4 33 2 4 4 33 3 4 4

33 13 124 4 33 14 124 4 33 11 124 4 33 12 124 4a

4

n133 1n 124 4

3 1n 124, 10

4 1 2 5 8

17 32 17 62 17 92 17 122 17 152 17 3 42 17 3 52

17 3 12 17 3 22 17 3 32a

5

n117 3n2

7 3n

a4

n133 1n 124 4a

5

n117 3n2

am

k1ck means c1 c2 c3 p cm

SummationNotation

If is an arithmetic sequence,then an expression of the form (sometimes written as

) is called an

arithmetic series.

aq

n1un

u1 u2 u3 p

u1, u2, u3, pNOTE

Using Calculators to Compute Sequences and Sums

Calculators can aid in computing sequences and sums of sequences. TheSEQ (or MAKELIST) feature on most calculators has the following syntax.

The last parameter, increment, is usually optional. When omitted, incre-ment defaults to 1. Refer to the Technology Tip about which menus containSEQ and SUM for different calculators.

The syntax for the SUM (or ) feature is

When start andend are omitted, the sum of the entire list is given.

Combining the two features of SUM and SEQ can produce sums ofsequences.

Example 8 Calculator Computation of a Sum

Use a calculator to display the first 8 terms of the sequence

and to compute the sum

Solution

Using the Technology Tip, enter which produces Fig-ure 1.3-4. Additional terms can be viewed by using the right arrow keyto scroll the display, as shown at right below.

SEQ17 3n, n, 1, 82,

a50

n17 3n.

un 7 3n

SUM(SEQ(expression, variable, begin, end))

SUM(list[, start, end]), where start and end are optional.

LIST

SEQ(expression, variable, begin, end, increment)

26 Chapter 1 Number Patterns

Figure 1.3-5

Figure 1.3-4

The first 8 terms of the sequence are and To compute the sum of the first 50 terms of the sequence, enter

Figure 1.3-5 shows the resulting display. There-

fore,

Partial Sums

Suppose is a sequence and k is a positive integer. The sum of the firstk terms of the sequence is called the kth partial sum of the sequence.

5un6

a50

n17 3n 3475.

SUM1SEQ17 3n, n, 1, 502 2.17.4, 1, 2, 5, 8, 11, 14,

Technology Tip

SEQ is in the OPSsubmenu of the TI

LIST menu and in theLIST submenu of the Casio OPTN menu.

SUM is in the MATHsubmenu of the TI LISTmenu. SUM is in the LISTsubmenu of the CasioOPTN menu.

Section 1.3 Arithmetic Sequences 27

Proof Let represent the kth partial sum Write theterms of the arithmetic sequence in two ways. In the first representationof repeatedly add d to the first term.

In the second representation of , repeatedly subtract d from the kth term.

If the two representations of are added, the multiples of d add to zeroand the following representation of is obtained.

Divide by 2.

The second formula is obtained by letting in the lastequation.

ku1 k 1k 12

2 d

Sk k2 1u1 uk2 k2 3u1 u1 1k 12d 4 k2 32u1 1k 12d 4

uk u1 1k 12d Sk

k2 1u1 uk2

2Sk 1u1 uk2 1u1 uk2 p 1u1 uk2 k1u1 uk2

Sk uk 3uk d 4 3uk 2d 4 p 3uk 1k 12d 4 Sk u1 3u1 d 4 3u1 2d 4 p 3u1 1k 12d 4

2SkSk

uk 3uk d 4 3uk 2d 4 p 3uk 1k 12d 4 Sk uk uk1 uk2 p u3 u2 u1

Sk

u1 3u1 d 4 3u1 2d 4 p 3u1 1k 12d 4 Sk u1 u2 u3 p uk2 uk1 uk

Sk ,

u1 u2 p uk.Sk

If { } is an arithmetic sequence with common difference d,then for each positive integer k, the kth partial sum can befound by using either of the following formulas.

1.

2. ak

n1un ku1

k(k 1)2 d

ak

n1un

k2 (u1 uk)

unPartial Sums of

an ArithmeticSequence

k terms

Calculator Exploration

Write the sum of the first 100 counting numbers. Then find a pat-tern to help find the sum by developing a formula using the termsin the sequence.

Example 9 Partial Sum of a Sequence

Find the 12th partial sum of the arithmetic sequence below.

Solution

First note that d, the common difference, is 5 and .

Using formula 1 from the box on page 27 yields the 12th partial sum.

Example 10 Partial Sum of a Sequence

Find the sum of all multiples of 3 from 3 to 333.

Solution

Note that the desired sum is the partial sum of the arithmetic sequenceThe sequence can be written in the form

where is the 111th term. The 111th partial sum of thesequence can be found by using formula 1 from the box on page 27 with

and

Example 11 Application of Partial Sums

If the starting salary for a job is $20,000 and you get a $2000 raise at thebeginning of each subsequent year, how much will you earn during thefirst ten years?

Solution

The yearly salary rates form an arithmetic sequence.

The tenth-year salary is found using and

20,000 9 120002 $38,000 u10 u1 110 12 d

d 2000.u1 20,000

20,000 22,000 24,000 26,000 p

a111

n1un

1112 13 3332 1112 13362 18,648

u111 333.u1 3,k 111,

333 3 111

3 1, 3 2, 3 3, 3 4, p ,

3, 6, 9, 12, p .

a12

n1un

122 18 472 234

8 11152 47 u12 u1 112 12 d

u1 8

8, 3, 2, 7, p

28 Chapter 1 Number Patterns

The ten-year total earnings are the tenth partial sum of the sequence.

$290,000 5158,0002

a10

n1un

102 1u1 u102 102 120,000 38,0002

Section 1.3 Arithmetic Sequences 29

Exercises 1.3

In Exercises 16, the first term, and the commondifference, d, of an arithmetic sequence are given. Findthe fifth term, the explicit form for the nth term, andsketch the graph of each sequence.

1. 2.

3. 4.

5. 6.

In Exercises 712, find the sum.

7. 8.

9. 10.

11. 12.

In Exercises 1318, find the kth partial sum of the arith-metic sequence { } with common difference d.

13.

14.

15.

16.

17.

18. k 10, u1 0, u10 30

k 6, u1 4, u6 14

k 9, u1 6, u9 24

k 7, u1 34, d

12

k 8, u1 23, d

43

k 6, u1 2, d 5

un

a31

n11300 1n 12 22a

36

n1512n 82

a75

n113n 12a

16

n112n 32

a4

i1

12ia

5

i13i

u1 p, d 15u1 10, d

12

u1 6, d 23u1 4, d

14

u1 4, d 5u1 5, d 2

u1, In Exercises 1924, show that the sequence is arith-metic and find its common difference.

19. 20.

21. 22.

23.

24.

In Exercises 2530, use the given information aboutthe arithmetic sequence with common difference d tofind and a formula for

25. 26.

27. 28.

29. 30.

In Exercises 3134, find the sum.

31. 32.

33. 34.

35. Find the sum of all the even integers from 2 to100.

36. Find the sum of all the integer multiples of 7 from7 to 700.

37. Find the sum of the first 200 positive integers.

38. Find the sum of the positive integers from 101 to200 (inclusive). Hint: Recall the sum from 1 to 100.Use it and Exercise 37.

a30

n1

4 6n3a

40

n1

n 36

a25

n1an4 5ba

20

n113n 42

u5 3, u9 18u5 0, u9 6

u7 6, u12 4u2 4, u6 32

u7 8, d 3u4 12, d 2

un.u5

52b 3nc6 1b, c constants25c 2n6 1c constant2

ep n2 fe5 3n

2 f

e4 n3f53 2n6

39. A business makes a $10,000 profit during its firstyear. If the yearly profit increases by $7500 in eachsubsequent year, what will the profit be in thetenth year? What will be the total profit for thefirst ten years?

40. If a mans starting annual salary is $15,000 and hereceives a $1000 increase to his annual salaryevery six months, what will he earn during thelast six months of the sixth year? How much willhe earn during the first six years?

41. A lecture hall has 6 seats in the first row, 8 in thesecond, 10 in the third, and so on, through row 12.Rows 12 through 20 (the last row) all have thesame number of seats. Find the number of seats inthe lecture hall.

30 Chapter 1 Number Patterns

42. A monument is constructed by laying a row of 60bricks at ground level. A second row, with twofewer bricks, is centered on that; a third row, withtwo fewer bricks, is centered on the second; andso on. The top row contains ten bricks