SGK TOÁN 12 của Nam Phi (English)

FHSST Authors The Free High School Science Texts: Textbooks for High School Students Studying the Sciences Mathematics Grade 12 Version 0.5 September 9, 2010 ii iii Copyright 2007 “Free High School Science Texts” Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later version published by the Free Software Foundation; with no Invariant Sections, no Front- Cover Texts, and no Back-Cover Texts. A copy of the license is included in the section entitled “GNU Free Documentation License”. STOP!!!! Did you notice the FREEDOMS we’ve granted you? Our copyright license is different! 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Thousands of hours went into making them and they are a gift to everyone in the education community. iv v FHSST Core Team Mark Horner ; Samuel Halliday ; Sarah Blyth ; Rory Adams ; Spencer Wheaton FHSST Editors Jaynie Padayachee ; Joanne Boulle ; Diana Mulcahy ; Annette Nell ; Ren´e Toerien ; Donovan Whitfield FHSST Contributors Sarah Abel ; Dr. Rory Adams ; Andrea Africa ; Ben Anhalt ; Prashant Arora ; Raymond Barbour ; Richard Baxter ; Tara Beckerling ; Tim van Beek ; Jennifer de Beyer ; Dr. Sarah Blyth ; Sebastian Bodenstein ; Martin Bongers ; Stephan Brandt ; Craig Brown ; Graeme Broster ; Deanne de Bude ; Richard Case ; Fanny Cherblanc ; Dr. Christine Chung ; Brett Cocks ; Andrew Craig ; Tim Crombie ; Dan Crytser ; Dr. Anne Dabrowski ; Laura Daniels ; Sean Dobbs ; Esmi Dreyer ; Matthew Duddy ; Fernando Durrell ; Dr . Dan Dwyer ; Frans van Eeden ; Alex Ellis ; Tom Ellis ; Giovanni Franzoni ; Ingrid von Glehn ; Tamara von Glehn ; Lindsay Glese ne r ; Kevin Godby ; Dr. Vanessa Godfrey ; Dr. Johan Gonzalez ; Hemant Gopal ; Dr. S te ph a ni e Gould ; Umeshree Govender ; Heather Gray ; Lynn Greeff ; Dr. Tom Gutierrez ; Brooke H aa g ; Kate Hadley ; Dr. Sam Halliday ; Ashee na Hanuman ; Dr Melanie Dymond Harper ; Dr. Nicholas Harrison ; Nei l Hart ; Nicholas Hatcher ; Dr. William P. Heal ; Pierre van Heerden ; Dr. Fritha Hennessy ; Millie Hilgart ; Chris Holdsworth ; Dr. Benne Holwerda ; Dr. M ark Horner ; Mfandaidza Hove ; Robert Hovden ; Jennifer Hsieh ; Clare Johnson ; Lu ke Jordan ; Tana Joseph ; Dr. Fabian J ut z ; Dr. Lutz Kampmann ; Paul Kim ; Dr. Jennifer Klay ; Lara Kruger ; Sihle Kubheka ; And re w Kubik ; Dr. Jannie Leach ; Dr. Marco van Leeuwen ; Dr. Tom Leinster ; Dr. Anton Machacek ; Dr. Komal Maheshwari ; Kosma von Maltitz ; Bryony Martin ; Nicole Masureik ; John Mathew ; Dr. Will Matthews ; JoEllen McBride ; Nikolai Meures ; Riana Meyer ; Filippo Miatto ; Jenny Miller ; Abdul Mirza ; Mapholo Modise ; Carla Moerdyk ; Asogan Moodaly ; Jothi Moodley ; David Myburgh ; Kamie Naidu ; Nolene Naidu ; Bridget Nash ; Tyrone Negus ; Thomas O’Donnell ; Dr. Markus Oldenburg ; Dr. Jaynie Padayachee ; Dave Pawson ; Nicolette Pekeur ; Sirika Pillay ; Jacques Plaut ; Andrea Prinsloo ; Joseph Raimondo ; Sanya Rajani ; Prof. Sergey Rakityansky ; Alastair Ramlakan ; Dr. M at in a J. Rassias ; Dr. Jocelyn Read ; Dr. Matthew Reece ; Razvan Remsing ; Laura Richter ; Max Richter ; Sean Riddle ; Jonath a n Reader ; Dr. David Roberts ; Evan Robinson ; Raoul Rontsch ; Dr. Andrew Rose ; Katie Ross ; Jeanne-Mari´e Roux ; Bianca Ruddy ; Katie Russell ; Steven Sam ; Nathaniel Schwartz ; Duncan Scott ; Helen Seals ; Ian Sherratt ; Dr. James Short ; Roger Sieloff ; Clare Slotow ; Bradley Smith ; Greg Solomon ; Dr. Andrew Stacey ; Dr. Jim Stasheff ; Mike Stay ; Mike Stringer ; Tim Teatro ; Ben Thompson ; Shen Tian ; Nicola du Toit ; Robert Torregrosa ; Jimmy Tseng ; Pieter Vergeer ; Helen Waugh ; Dr. Dawn Webber ; Michelle Wen ; Neels van der Westhuizen ; Dr. Alexander Wetzler ; Dr. Spencer Wheaton ; Vivian White ; Dr. Gerald Wigger ; Harry Wiggins ; He ath e r Williams ; Wendy Williams ; Julie Wilson ; Timothy Wilson ; Andrew Wood ; Emma Wormauld ; Dr. Sahal Yacoob ; Jean Youssef ; Ewald Zietsman Contributors and editors have made a sincere effort to produce an accurate and useful resource. Should you have suggestions, find mistakes or be prepared to donate material for inclusion, please don’t hesitate to conta c t us. We intend to work with all who are willing to help make this a continuously evolving resource! www.fhsst.org vi Contents 1 Introduction to Book 1 1.1 The Language of Mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2 Logarithms - Grade 12 3 2.1 Definition of Logarithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.2 Logarithm Bases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.3 Laws of Logarithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.4 Logarithm Law 1: log a 1 = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.5 Logarithm Law 2: log a (a) = 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.6 Logarithm Law 3: log a (x · y) = log a (x) + log a (y) . . . . . . . . . . . . . . . . . 6 2.7 Logarithm Law 4: log a  x y  = log a (x) − log a (y) . . . . . . . . . . . . . . . . . 7 2.8 Logarithm Law 5: log a (x b ) = b log a (x) . . . . . . . . . . . . . . . . . . . . . . . 8 2.9 Logarithm Law 6: log a ( b √ x) = log a (x) b . . . . . . . . . . . . . . . . . . . . . . . 8 2.10 Solving simple log equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.10.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.11 Logarithmic applications in the Real World . . . . . . . . . . . . . . . . . . . . . 12 2.11.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.12 End of Chapter Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 3 Sequences and Series - Grade 12 15 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 3.2 Arithmetic Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 3.2.1 General Equation for the n th -term of an Arithmetic Sequence . . . . . . 16 3.3 Geometric Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 3.3.1 Example - A Flu Epidemic . . . . . . . . . . . . . . . . . . . . . . . . . 17 3.3.2 General Equation for the n th -term of a Geometric Sequence . . . . . . . 19 3.3.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 3.4 Recursive Formulae for S eq u en ce s . . . . . . . . . . . . . . . . . . . . . . . . . 20 3.5 Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 3.5.1 Some Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 3.5.2 Sigma Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 3.6 Finite Arithmetic Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 3.6.1 General Formula for a Finite Arithmetic Series . . . . . . . . . . . . . . . 25 vii CONTENTS CONTENTS 3.6.2 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 3.7 Finite Squared Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 3.8 Finite Geometric Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 3.8.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 3.9 Infinite Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 3.9.1 Infinite Geometric Series . . . . . . . . . . . . . . . . . . . . . . . . . . 29 3.9.2 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 3.10 End of Chapter Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 4 Finance - Grade 12 35 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 4.2 Finding the Length of the Investment or Loan . . . . . . . . . . . . . . . . . . . 35 4.3 A Series of Payments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 4.3.1 Sequences and Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 4.3.2 Present Values of a series of Payments . . . . . . . . . . . . . . . . . . . 37 4.3.3 Future Value of a series of Payments . . . . . . . . . . . . . . . . . . . . 42 4.3.4 Exercises - Present and Future Values . . . . . . . . . . . . . . . . . . . 43 4.4 Investments and Loans . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 4.4.1 Loan Schedules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 4.4.2 Exercises - Investments and Loa ns . . . . . . . . . . . . . . . . . . . . . 46 4.4.3 Calculating Capital Outstanding . . . . . . . . . . . . . . . . . . . . . . 46 4.5 Formulae Sheet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 4.5.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 4.5.2 Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 4.6 End of Chapter Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 5 Factorising Cubic Polynomials - Grade 12 49 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 5.2 The Factor Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 5.3 Factorisation of Cub ic Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . 50 5.4 Exercises - Using Factor Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 52 5.5 Solving Cubic Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 5.5.1 Exercises - Solving of Cubic Equations . . . . . . . . . . . . . . . . . . . 54 5.6 End of Chapter Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 6 Functions and Graphs - Grade 12 57 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 6.2 Definition of a Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 6.2.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 6.3 Notation used for Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 6.4 Graphs of Inverse Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 6.4.1 Inverse Function of y = ax + q . . . . . . . . . . . . . . . . . . . . . . . 59 6.4.2 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 viii CONTENTS CONTENTS 6.4.3 Inverse Function of y = ax 2 . . . . . . . . . . . . . . . . . . . . . . . . 61 6.4.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 6.4.5 Inverse Function of y = a x . . . . . . . . . . . . . . . . . . . . . . . . . 62 6.4.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 6.5 End of Chapter Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 7 Differential Calculus - Grade 12 65 7.1 Why do I have to learn this stuff? . . . . . . . . . . . . . . . . . . . . . . . . . 65 7.2 Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 7.2.1 A Tale of Achilles and the Tortoise . . . . . . . . . . . . . . . . . . . . . 66 7.2.2 Sequences, Series and Functions . . . . . . . . . . . . . . . . . . . . . . 67 7.2.3 Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 7.2.4 Average Gradient and Gradient at a Point . . . . . . . . . . . . . . . . . 71 7.3 Differentiation from First Principles . . . . . . . . . . . . . . . . . . . . . . . . . 75 7.4 Rules of Differentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 7.4.1 Summary of Differentiation Rules . . . . . . . . . . . . . . . . . . . . . . 77 7.5 Applying Differentiation to Draw Graphs . . . . . . . . . . . . . . . . . . . . . . 78 7.5.1 Finding Equations of Tangents to Curves . . . . . . . . . . . . . . . . . 78 7.5.2 Curve Sketching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 7.5.3 Local minimum, Local maximum and Point of Inflextion . . . . . . . . . 84 7.6 Using Differential Calculus to Solve Problems . . . . . . . . . . . . . . . . . . . 85 7.6.1 Rate of Change problems . . . . . . . . . . . . . . . . . . . . . . . . . . 89 7.7 End of Chapter Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 8 Linear Programming - Grade 12 95 8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 8.2 Terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 8.2.1 Feasible Region and Points . . . . . . . . . . . . . . . . . . . . . . . . . 95 8.3 Linear P rogrammi n g and the Feasible Region . . . . . . . . . . . . . . . . . . . 96 8.4 End of Chapter Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 9 Geometry - Grade 12 105 9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 9.2 Circle Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 9.2.1 Terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 9.2.2 Axioms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 9.2.3 Theorems of the Geometry of Circles . . . . . . . . . . . . . . . . . . . . 106 9.3 Co-ordinate Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 9.3.1 Equation of a Circle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 9.3.2 Equation of a Tangent to a Circle at a Point on the Circle . . . . . . . . 125 9.4 Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 9.4.1 Rotation of a Point about an angle θ . . . . . . . . . . . . . . . . . . . . 127 9.4.2 Characteristics of Transformations . . . . . . . . . . . . . . . . . . . . . 129 9.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 ix CONTENTS CONTENTS 10 Trigonometry - Grade 12 133 10.1 Compound Angle Identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 10.1.1 Derivation of sin(α + β) . . . . . . . . . . . . . . . . . . . . . . . . . . 133 10.1.2 Derivation of sin(α − β) . . . . . . . . . . . . . . . . . . . . . . . . . . 134 10.1.3 Derivation of cos(α + β) . . . . . . . . . . . . . . . . . . . . . . . . . . 134 10.1.4 Derivation of cos(α − β) . . . . . . . . . . . . . . . . . . . . . . . . . . 135 10.1.5 Derivation of sin 2α . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 10.1.6 Derivation of cos 2α . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 10.1.7 Problem-solving Strategy for Identities . . . . . . . . . . . . . . . . . . . 136 10.2 Applications of Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . . 138 10.2.1 Problems in Two Dimensions . . . . . . . . . . . . . . . . . . . . . . . . 138 10.2.2 Problems in 3 dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . 140 10.3 Other Geometries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142 10.3.1 Taxicab Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142 10.3.2 Manhattan distance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142 10.4 End of Chapter Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 11 Statistics - Grade 1 2 145 11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 11.2 A Normal Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 11.3 Extracting a Sample Population . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 11.4 Function Fitting and Regression Analysis . . . . . . . . . . . . . . . . . . . . . . 148 11.4.1 The Method of Least Squares . . . . . . . . . . . . . . . . . . . . . . . 150 11.4.2 Using a calculator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 11.4.3 Correlation coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . 154 11.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155 12 Combinations and Permutations - Grade 12 159 12.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 12.2 Counting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 12.2.1 Making a List . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 12.2.2 Tree Diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160 12.3 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160 12.3.1 The Factorial Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . 160 12.4 The Fundamental Counting Principle . . . . . . . . . . . . . . . . . . . . . . . . 160 12.5 Combinations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161 12.5.1 Counting Combinations . . . . . . . . . . . . . . . . . . . . . . . . . . . 161 12.5.2 Combinatorics and Probability . . . . . . . . . . . . . . . . . . . . . . . 162 12.6 Permutations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162 12.6.1 Counting Permutations . . . . . . . . . . . . . . . . . . . . . . . . . . . 163 12.7 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164 12.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166 x [...]... logb a (2.3) where b is any base you find convenient Normally a and b are known, therefore logb a is normally a known, if irrational, number For example, change log2 12 in base 10 is: log2 12 = 4 log10 12 log10 2 CHAPTER 2 LOGARITHMS - GRADE 12 2.3 Activity :: Change of Base : Change the following to the indicated base: 1 log2 (4) to base 8 2 log10 (14) to base 2 3 log16 (4) to base 10 4 logx (8) to base... LOGARITHMS - GRADE 12 2.9 √ Activity :: Logarithm Law 6: loga ( b x) = √ 1 log2 ( 4 8) √ 2 log8 ( 10 10) √ 3 log16 ( y x) √ 4 logz ( x y) √ 5 logx ( 2x y) loga (x) b : Simplify the following: The final answer doesn’t have to look simple Worked Example 1: Simplification of Logs Question: Simplify, without use of a calculator: 3 log 3 + log 125 Answer Step 1 : Try to write any quantities as exponents 125 can be... log 3 = n = n = n (1,05) 2 n × log 1,05 (using law 5) 2 2 log 3 ÷ log 1,05 45,034 Step 3 : Final answer So it will take approximately 45 years for the population to triple in size 12 CHAPTER 2 LOGARITHMS - GRADE 12 2.11.1 2 .12 Exercises 1 The population of a certain bacteria is expected to grow exponentially at a rate of 15 % every hour If the initial population is 5 000, how long will it take for the... for at least 8 years 2 .12 End of Chapter Exercises 1 Show that x y loga 2 Show that loga = loga (x) − loga (y) √ b x = loga (x) b 3 Without using a calculator show that: log 5 32 75 − 2 log + log = log 2 16 9 243 4 Given that 5n = x and n = log2 y (a) Write y in terms of n (b) Express log8 4y in terms of n (c) Express 50n+1 in terms of x and y 13 2 .12 CHAPTER 2 LOGARITHMS - GRADE 12 5 Simplify, without... 5 − 9−1 √ 5 1 2 + log3 92 ,12 6 Simplify to a single number, without use of a calculator: log 32 − log 8 log 8 (b) log 3 − log 0,3 (a) log5 125 + 7 Given: log3 6 = a and log6 5 = b (a) Express log3 2 in terms of a (b) Hence, or otherwise, find log3 10 in terms of a and b 8 Given: Prove: pq k = qp−1 k = 1 − 2 logq p 9 Evaluate without using a calculator: (log7 49)5 + log5 1 125 − 13 log9 1 10 If log 5... 7: Question: Answer Logs in Compound InterestI have R12 000 to invest I need the money to grow to at least R30 000 If it is invested at a compound interest rate of 13% per annum, for how long (in full years) does my investment need to grow ? Step 1 : The formula to use A = P (1 + i)n Step 2 : Substitute and solve for n 30000 1,13n > n log(1,13) 120 00(1 + 0,13)n 5 2 log(2,5) < > n > n > log(2,5) ÷ log(1,13)... to get the following sequences and write out the next 5 terms 1 2,6,10,14,18,22, 2 −5, − 3, − 1,1,3, 3 1,4,7,10,13,16, 4 −1,10,21,32,43,54, 5 3,0, − 3, − 6, − 9, − 12, 15 3.2 CHAPTER 3 SEQUENCES AND SERIES - GRADE 12 3.2.1 General Equation for the nth -term of an Arithmetic Sequence More formally, the number we start out with is called a1 (the first term), and the difference between each... Activity :: Common Factor of Geometric Sequence : Determine the common factor for the following geometric sequences: 1 5, 10, 20, 40, 80, 2 1 1 1 2,4,8, 3 7, 28, 112, 448, 4 2, 6, 18, 54, 18 CHAPTER 3 SEQUENCES AND SERIES - GRADE 12 3.3 5 −3, 30, −300, 3000, 3.3.2 General Equation for the nth -term of a Geometric Sequence From the above example we know a1 = 2 and r = 2, and we have seen from... Determine the formula for the following geometric sequences: 1 5, 10, 20, 40, 80, 2 1 1 1 2,4,8, 3 7, 28, 112, 448, 4 2, 6, 18, 54, 5 −3, 30, −300, 3000, 3.3.3 Exercises 1 What is the important characteristic of an arithmetic sequence? 19 3.4 CHAPTER 3 SEQUENCES AND SERIES - GRADE 12 2 Write down how you would go about finding the formula for the nth term of an arithmetic sequence? 3 A single... 5 = 15 i=1 Examples 1 6 = 21 + 22 + 23 + 24 + 25 + 26 = = 2i 2 + 4 + 8 + 16 + 32 + 64 126 i=1 2 10 (3xi ) = 3x3 + 3x4 + + 3x9 + 3x10 i=3 for any value x Some Basic Rules for Sigma Notation 1 Given two sequences, ai and bi , n n n (ai + bi ) i=1 i=1 22 bi ai + = i=1 (3.16) CHAPTER 3 SEQUENCES AND SERIES - GRADE 12 3.6 2 For any constant c that is not dependent on the index i, n c · ai = c · a1 + . . . . . . . . . . . 155 12 Combinations and Permutations - Grade 12 159 12. 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 12. 2 Counting . . . . known, if irrational, number. For example, change log 2 12 in base 10 is: log 2 12 = log 10 12 log 10 2 4 CHAPTER 2. LOGARITHMS - GRADE 12 2.3 Activity :: Change of Base : Change the following. . . . . 127 9.4.1 Rotation of a Point about an angle θ . . . . . . . . . . . . . . . . . . . . 127 9.4.2 Characteristics of Transformations . . . . . . . . . . . . . . . . . . . . . 129 9.5 Exercises