Clever Keeping Maths Simple

Clever Keeping Maths Simple Grade Learner s Book J Aird L du Toit I Harrison C van Dun J van Dun

Clever Keeping Maths Simple Grade Learner s Book J Aird, L du Toit, I Harrison, C van Dun and J van Dun, 0 Illustrations and design: Macmillan South Africa (Pt) Ltd, 0 All rights reserved. No part of this publication ma be reproduced, stored in a retrieval sstem, or transmitted in an form or b an means, electronic, photocoping, recording, or otherwise, without the prior written permission of the copright holder or in accordance with the provisions of the Copright Act, 978 (as amended). An person who commits an unauthorised act in relation to this publication ma be liable for criminal prosecution and civil claims for damages. First published 0 5 7 6 4 0 4 6 8 0 9 7 5 Published b Macmillan South Africa (Pt) Ltd Private Bag X9 Northlands 6 Gauteng South Africa Tpeset b Ink Design Cover image b Tamara Joubert Cover design b Future Prepress Illustrations b MPS and Geoff Walton Photographs b: AAI Fotostock: page 5 ISBN: 978480644 e-isbn: 9784804658 WIP: 04K000 It is illegal to photocop an page of this book without written permission from the publishers. The publishers have made ever effort to trace the copright holders. If the have inadvertentl overlooked an, the will be pleased to make the necessar arrangements at the first opportunit. The publishers would also like to thank those organisations and individuals we have alread approached and from whom we are anticipating permission.

Contents Chapter Patterns, sequences and series... Introduction... Determine whether a sequence is arithmetic or geometric... The general term of a sequence... 4 A comparison of arithmetic and geometric progressions... 7 Arithmetic and geometric means... 0 Sigma notation... Series... 5 The sum of an arithmetic series... 5 The sum of a geometric series... 6 Mied problems... 0 Applications of arithmetic and geometric sequences and series... Infinite series... 6 Summar... 7 Chapter Functions and inverse functions... 40 Revising functions dealt with in Grade 0 and... 40 Functions... 47 Inverse functions... 49 Chapter Eponential and logarithmic functions... 6 Eponential functions... 6 Logarithmic functions... 64 Definition of a logarithm... 66 Logarithmic laws... 68 Equations involving logarithms... 7 Eponential and logarithmic functions... 7 Chapter 4 Finance, growth and deca... 8 Grade summar... 8 Future value annuities... 86 The formula for future value annuities... 89 Loans and loan repaments... 9 The present value formula for annuities... 94 Deferred annuities... 98 Calculating the time period, n... 0 Chapter 5 Compound angles... 08 Proofs of the compound angle formulae... 08 Evaluating compound angle ratios using special angles... Using compound angle formulae to prove identities... 5 Double angle formulae... 6 More comple eamples... 9 Identities involving double angles... 0

Chapter 6 Trigonometric equations... 4 Revision of Grade trigonometric equations... 4 Grade trigonometric equations... 7 Mied equations... 0 Chapter 7 Solving problems using trigonometr... 4 The basic requirements for solving triangles... 4 The sine, cosine and area rules... 5 Triangles in three dimensions... 44 Identities using the sine, cosine and area rules... 55 Chapter 8 Polnomial functions... 6 Revision... 6 Using long division to find a remainder... 6 The Remainder Theorem... 65 The Factor Theorem... 67 Determining the quotient and remainder b snthetic division... 67 Factorising... 69 Factorising epressions in two variables... 7 Using the Remainder Theorem to determine unknown coefficients... 7 Solving cubic equations using the Remainder Theorem... 7 Chapter 9 Differential calculus... 76 The concept of a limit... 77 Introduction to calculus... 80 Average gradient... 80 Calculating the derivative from first principles... 85 The gradient and equation of a tangent to a curve... 9 Equations of tangents to a curve... 94 The cubic graph... 98 Problems involving maima and minima... The derivative as a wa to measure the rate of change... 0 Chapter 0 Analtical geometr... 8 Revision of formulae from Grade... 8 The equation of a circle... 9 Working with circles... 4 The equation of a tangent to a circle... 48 Summar... 5

Chapter Euclidean geometr... 65 Revision: similarit of polgons... 65 Revision... 66 Proportionalit in triangles... 7 Similar triangles... 8 Equiangular triangles are similar... 8 Triangles with sides in proportion are similar... 84 The Theorem of Pthagoras and similar triangles... 97 Summar... Chapter Statistics (regression and correlation)... 5 Smmetric and skewed data (revision)... 6 Bivariate data: scatter plots... 5 Bivariate data: scatter plots, regression and correlation... 9 Using a calculator to do regression calculations... Interpolation and etrapolation... Summar... 47 Chapter Counting and probabilit... 48 Revision... 48 Solving problems... 50 The fundamental counting principle... 60 Factorial notation... 66 Special cases of the fundamental counting principle... 68 Permutations and combinations... 7 Use the fundamental counting principle to solve probabilit problems... 74 Summar... 79 Chapter 4 Revision... 8 Chapter Patterns, sequences and series... 8 Chapter Functions and inverse functions... 84 Chapter Eponential and logarithmic functions... 87 Chapter 4 Financial mathematics... 9 Chapters 5, 6 and 7 Trigonometr... 9 Chapter 8 Functions: polnomials... 97 Chapter 9 Differential calculus... 97 Chapter 0 Analtical geometr... 400 Chapter Euclidean geometr... 40 Chapter Statistics (regression and correlation)... 407 Chapter Counting and probabilit... 4

Sample papers...44 Mid-ear Paper...44 Mid-ear Paper...48 End of ear Paper...4 End of ear Paper...47 Memorandum for Mid-ear Paper...4 Memorandum for Mid-ear Paper...47 Memorandum for End of ear Paper...44 Memorandum for End of ear Paper...450 Answers to eercises... 456 Glossar of mathematical terms... 50

Chapter Patterns, sequences and series In this chapter ou will: learn about arithmetic and geometric sequences write different series in sigma notation derive and use the formulae for the sum of arithmetic and geometric series. A sequence is an ordered set of numbers. Introduction A progression is a sequence in which we can obtain the value of an element based on the values of the preceding elements. The following table shows the first four terms of two different sequences. We refer to each term in a sequence using the notation T n where n represents the position of the term. Both sequences are progressions. Term Term Term Term 4 T T T T 4 Sequence 7 4 8 Sequence 4 8 6 A series is the sum of the elements of a sequence. For eample, if we add the values of the second sequence in the previous table, we have the series + 4 + 8 + 6 +. Arithmetic progressions (AP) An arithmetic progression (AP) is a sequence in which each term after the first term is formed b adding a constant value (d) to the preceding term. Geometric progressions (GP) A geometric progression (GP) is a sequence in which each term after the first term is formed b multipling the preceding term b a constant ratio (r).

Eample Eample T T T T 4 T 5 5 8 4 = + = 5 + = 8 + = + This is an arithmetic sequence with d = Eample T T T T 4 6 8 54 = = 6 = 8 This is a geometric sequence with r = Eample T T T T 4 T 5 6 8 4 0 = 6 4 = 4 = 8 4 = 4 4 This is an arithmetic sequence with d = 4 Eample If T n = 4 n, determine the sequence. T = 4 () = 4 = T = 4 () = 4 4 = 0 T = 4 () = 4 6 = the sequence is ; 0; ; T T T T 4 6 8 4 = 6 ( ) = 8 ( ) = 4 ( This is a geometric sequence with r = Eample If T n =. n, determine the sequence. T =. =. 0 =. = T =. =. =. = 6 T =. =. =. 4 = the sequence is ; 6; ; _ ) Determine whether a sequence is arithmetic or geometric Given a sequence, we can use a formula to test if a sequence is arithmetic, geometric or neither. Arithmetic progressions (AP) To test whether a sequence is arithmetic, use the formula : T T = T T = d Geometric progressions (GP) To test whether a sequence is geometric, use the formula : T T = T T = r

Eample Determine whether the following sequence is arithmetic, geometric or neither. ; 4; 6; 8; 0; Eample Determine whether the following sequence is arithmetic, geometric or neither. ; 6; 8; 54; T T T T 4 T 5 4 6 8 0 T T = 6 4 = T T = 4 = This is an arithmetic sequence with d =. T T T T 4 6 8 54 T T = 8 6 = T T = 6 = This is a geometric sequence with r =. Eercise.. Determine whether the following sequences are arithmetic, geometric or neither: a) 5; 8; ; 4; 7; b) 5; 0; 0; 40; 80; c) ; 4; 9; 6; 5; d) ; 6; 8; 4; ; e) 6; 0; 4; 8; ; f) ; 7; ; ; 8; g) ; 4 ; 5 6 ; 7 8 ; 9 ; 0 h) ; ; 9; 7; 8; i) 7; 4; ; ; 5; j) ; 8; 7; 64; 5; k) ; ; 9 ; 7 4 ; 8 8 ; l) ; 7 ; 5; ; 8; 9 ; m) ; 9; 7; 8; 4; n) ; ; ; 9 ; 7 ; o) ; ; ; ; 5; 8; ;. Given the general term of the sequence: i) Determine the first five terms of the sequence. ii) State whether the sequence is arithmetic, geometric or neither. a) T n = n + b) T k = k c) T n = n d) T n = n e) T k = k + 5 f) T n = 5n g) T n = 7 n h) T k =. k i) n T n = j) T n = 4. n n +

The general term of a sequence In this section, we eplain how to find the general term of a sequence. Eercise. Fill in the spaces: Arithmetic progressions (AP) Given the sequence: ; 5; 8; ; 4; 7; with d = Term: general form T = a T = 5 = + : a + d T = 8 = + + = + (): a + d T 4 = = + + + = + (): T 5 = 4 = + + + + = + 4(): T 6 = 7 = + + + + + = + 5(): T 0 = T = T n = Geometric progressions (GP) Given the sequence: ; ; 4; 8; 6; ; 64. with r = Term: general form T = : a T = = : ar T = 4 = = : ar T 4 = 8 = = : T 5 = 6 = = 4 : T 6 = = = 5 : T 0 = T = T n = From the previous eercise, we found that: the general term of an arithmetic sequence is T n = a + (n )d the general term of a geometric sequence is T n = a. r n 4

Arithmetic progressions (AP) Eample Determine the th term of the sequence ; 7; ; We first need to determine whether the sequence is arithmetic or geometric: T T = 7 = 4 T T = 7 = 4 the sequence is arithmetic with d = 4 Since T n = a + (n )d, the th term will be: T = a + ( )d = a + d We know that a =, d = 4 and n = : T = a + d = + (4) = 47 Eample Determine the nth term of the sequence ; 6; ; 6; T T T... T n a a + d a + d a + (n )d 6 a =, d = 5 and T n = a + (n )d: T n = a + (n )d T n = + (n )5 T n = + 5n 5 T n = 5n 4 Geometric progressions (GP) Eample Determine the eighth term of the sequence ; ; ; Determine whether the sequence is arithmetic or geometric: T T = = T T = the sequence is geometric with r = Since T n = a. r n, the eighth term will be: T 8 = a. r 8 = ar 7 We know that a =, r = and n = 8: T 8 = ar 7 = ( ) 7 = 8 = 64 Eample Determine T n for the sequence ; 9; 7; T T T T n a ar ar ar n 9 7 a =, r = and T n = ar n : T n = ar n = ()n = n = + n = n 5

Eample Determine which term is equal to 06 in the following arithmetic progression: 8; ; 4; T T T T n a a + d a + d a + (n )d 8 4 06 a = 8, d = 6, n =? and T n = 06: T n = a + (n )d 06 = 8 + (n )( 6) 06 = 8 6n + 6 6n = 4 + 06 6n = 0 6n 6 = 0 6 n = 0 the 0th term is equal to 06. Eample Determine which term is equal to 8 79 in the following geometric progression: ; ; 4 ; T T T T n a ar ar ar n 4 8 _, n =? and T = 8 n 79 a =, r = T n = ar n 8 = 79 ( _ ) n _ 8 = _ 79 ( _ ) n 8 = 87 ( _ ) n ( _ ) n 8 = 79 Divide both sides b to isolate the power To solve, we need 87 8 to write as a 87 power of _ ( ) n = ( ) 7 n = 7 Equate the eponents n = 8 the eighth term is equal to 8 79 Eercise.. Determine the required term in each sequence: a) T of ; 7; ; b) T 7 of 6; ; 4; c) T 9 of ; ; 9; 7; d) T 5 of 0; ; 4; e) T of 7; 0; ; f) T of ; 8; 4; g) T 7 of ; ; 48; h) T 4 of 4; 4; 4; i) T 8 of ; 6; 9; j) T 0 of 5; 75; 75;. Determine the number of terms in each of the following sequences: a) ; 7; ; 47 b) ; 6; ; 96 c) 5; 8; ; 4 d) 0; ; 4; 0 e) 5; 0; 0; 5 0 f) 4; ; 6; 08; 8 748 g) 4; 7 ; ; 5 ; h) 4; ; ; ; 56 7 i) 7; 8 ; 7 8 ; 7 j) 0; 9 80 4 ; 8 ; 6

A comparison of arithmetic and geometric progressions Arithmetic progressions (AP) Eample In an arithmetic sequence, T = 4 and T 0 =. Determine:. the sequence.. the 5th term.. T = 4 and T 0 = a = 4 T n = a + (n )d = 4 + (0 )d T 0 = 4 + 9d General form 9d = 4 9d = 7 7 d = 9 = the sequence is 4; 7; 0;. a = 4 and d = T 5 = a + 4d = 4 + 4 = 46 The 5th term is 46. Geometric progressions (GP) Eample In a geometric sequence, T = 4 and T 0 = 8. Determine:. the sequence.. the 5th term.. T = 4 and T 0 = 8 a = 4 ar 9 = General form 8 To solve for r, substitute a = 4 into the equation ar 9 = 8 4r 9 = 8 r 9 = 5 Divide both sides b 4 r 9 = ( _ ) 9 r = the sequence is 4; ; ;. a = 4 r = _ T 5 = ar 4 = 4( _ ) 4 = 4 096 7

Eample In an arithmetic sequence, T = and T 8 =. Determine the first term and the common difference. T = and T 8 = We need to solve these two equations simultaneousl. We can do this using the substitution method or b elimination. Here, we use the elimination method. a + d = a + 7d = a + 7d = (a + d = ) 5d = 5 d = 5 Therefore: d = 5 a + (5) = Substitute d = 5 into to solve for a a + 0 = a = The first term is. The common difference is 5. Eample In a geometric sequence, T 4 = and T 6 =. Determine the second term. T 4 = _ T = _ 6 ar = _ ar5 = _ Divide equation b equation to eliminate a. Then, solve for r. ar5 ar = r = r = 9 4 r = ± Since the value of r is squared, there will be two solutions to this equation Since there are two values of r, there will be two different sequences. To find the value of a, substitute the value of r into equation : If r = _ If r = _ a ( _ ) = _ a ( _ ) = _ a( 7 8 ) = _ 7 a( 8 ) = _ a( 7 8 ) 8 8 7 a( 7 7 8 ) 8 8 7 7 a = 8 7 a = 8 7 6 a = 8 a = 6 8 T = ar T = ar 6 = _ 8 8 = 7 6 = _ 8 8 = 7 8

Eample If ( + ); ( + 4); ( + 4); is an arithmetic sequence, calculate the value of. T T = T T ( + 4) ( + 4) = ( + 4) ( + ) + 4 4 = + 4 = Therefore: T = + = 4 T = + 4 = 7 T = + 4 = 0 The sequence is 4; 7; 0; Eample If ( + ); ( + ); ( + ) is a geometric progression, calculate the value of. T = T T T + + = + + ( + )( + ) = ( + )( + ) + 5 + = + 6 + 9 6 = 0 ( )( + ) = 0 = or = If =, then the sequence is 4; 6; 9; If =, then the sequence is ; ; ; Eercise.4. Determine T 0 of the following arithmetic sequences: a) a = 4 and d = b) T = 0 and T 6 = 60 c) T 5 = 8 and d = d) T = 4 and T 5 = 0. Determine T 0 of the following geometric sequences: a) a = 4 and r = b) T = 0 and T 6 = 60 4 c) T 5 = 8 and r = _ d) T = 4 and T 5 = 4 8. Calculate the following terms: a) T of the arithmetic progression, if T 5 = and T 0 = 4. b) T 5 of the geometric progression, if T = 4 and T = 8. 8 c) T 4 of the geometric progression, if T 6 = 64 and T 0 = 04. d) T 7 of the arithmetic progression, if T 5 = and T 8 =. 4. Give the first three terms of an arithmetic sequence in which is the seventh term and the th term is. 5. Which term of the sequence ; 7; ; is? 6. If ; ; 4 + are three successive terms of a geometric sequence: a) Calculate the value of. b) Determine the sequence. c) Determine the 9th term of the sequence. d) Which term of the sequence will be equal to 4 74? 7. The first two terms of an arithmetic sequence are m and n, respectivel. Calculate the 0th term. 9

8. If ; + ; 5 + are the first three terms of an arithmetic sequence, calculate the value of. 9. Determine the geometric progression in which T 4 = 4 and the common ratio is. 0. Find which term of the sequence 9; 5; ; is 45.. The numbers 4; ; form an arithmetic sequence. The numbers ; ; 8 form a geometric sequence. Calculate the values of and.. Given the sequence ; 8; 4; 0; a) Determine the 50th term. b) Which term will be equal to 50?. The following is an arithmetic sequence: 6 + ; 4 + 7; + 4; a) Calculate the value of. b) Write down the value of: i) the first term of the sequence ii) the common difference iii) the fifth term. 4. a) Determine T 5 of the sequence + ; + 4; + 7; b) Which term of the sequence is equal to + 6? 5. A bo is repaing a debt to a friend. He pas R0 in the first week, R5 in the second week, R0 in the third week, and so on. If he finishes paing after the eighth week, how much was his last pament? Arithmetic and geometric means Arithmetic mean (AM) If a; ; b is an arithmetic progression then is the arithmetic mean of a and b. T T = T T b = a = a + b a + b = the arithmetic mean is a + b Eample Calculate the arithmetic mean of and. + AM = = 4 = 7 Therefore, ; 7; ; is an arithmetic sequence. Geometric mean (GM) If a; ; b is a geometric progression then is the geometric mean of a and b. T T = T T b = (a,, b 0) a = ab = + ab B definition, the geometric mean is ab. Eample Calculate the geometric mean of and 8. GM = 8 = 6 = 4 Therefore ; 4; 8; is a geometric sequence. 0

Eample Insert three arithmetic means between and 0. This means we need to insert three numbers between and 0 such that the numbers form an arithmetic sequence. T T T T 4 T 5 a a + 4d 0 So T = a = T 5 = 0 a + 4d = 0 + 4d = 0 4d = d = the sequence is: ; ; 4; 7; 0 Eample Insert two geometric means between and. This means we need to insert two numbers between and such that the numbers form a geometric sequence. T T T T 4 a ar So T = T 4 = a = ar = ( )r = r = r = the sequence is ; ; ; Eercise.5. Determine the arithmetic mean between a) 4 and 8 b) and c) 8 and d),7 and 4,5. Determine the geometric mean of a) and b) 5 and 0 c) and 50 d) 4 and 6. Insert four arithmetic means between 4 and 9. 4. Insert three geometric means between and 4. 4 Sigma notation The smbol is the Greek smbol sigma. We use this smbol to find the sum of a sequence. When we add the terms in a sequence, we call this a series. For eample, if we have a sequence 5; 8; ; 4; 7, then the series is 5 + 8 + + 4 + 7.

Eample 5 Evaluate: (n + ) n = 5 n = (n + )means that we need to find the sum of a sequence. To find the terms of the sequence, we first substitute n = into the general term of n +. Then, we substitute n =, n =, and so on, until we reach n = 5. So, the number below the sigma smbol tells us where to start (n = ), and the number above the sigma smbol tells us where to stop (n = 5). 5 So (n + ) = (. + ) + (. + ) + (. + ) + (. 4 + ) + (. 5 + ) n = = 5 + 8 + + 4 + 7 = 55 Eample 6 Evaluate: (. r ) r = 0 In this case, we start with r = 0 and stop when r = 6. In other words, we begin b substituting r = 0 into the general term. r and continue to r = 6. 6 r = 0 (. r ) =. 0 +. +. +. +. 4 +. 5 +. 6 Eample =. +. +. 9 +. 7 +. 8 +. 4 +. 79 = + 6 + 8 + 54 + 6 + 486 + 458 = 86 0 Evaluate: ( i + 5) i = In this case, we start b substituting i = into the general term i + 5, and continue to i = 0: 0 i = ( i + 5)= (. + 5) + (. 4 + 5) + (. 5 + 5) + (. 6 + 5) + (. 7 + 5) + (. 8 + 5) + (. 9 + 5) + (. 0 + 5) = ( 6 + 5) + ( 8 + 5) + ( 0 + 5) + ( + 5) + ( 4 + 5) + ( 6 + 5) + ( 8 + 5) + ( 0 + 5) = 5 7 9 5 = 64

Number of terms = top number bottom number + 5 In Eample, (n + )the number of terms is 5 + = 5. n = 6 In Eample, (. r ) the number of terms is 6 0 + = 7. r = 0 0 In Eample, ( i + 5)the number of terms is 0 + = 8. b i = So if T n the number of terms in the series will be b a +. n = a Also, note that each eample used different variables (n, r and i). Eample 4 6 Evaluate: 5 n = We know that there are 6 + = 6 terms in this series. Here, the general term is a constant, so we write the series as follows: 6 n = 5 = 5 + 5 + 5 + 5 + 5 + 5 = 6 5 = 0 Writing a series in sigma notation Eample Write the following series in sigma notation: 5 + 8 + + 4 + 7 First, we need to determine the general term, T n, for the series. The series is an arithmetic series with a = 5 and d =. Therefore: T n = a + (n )d = 5 + (n ) = 5 + n = n + Now we write in the formula after the sigma smbol:... n =... (n + ) Net, we need to determine the numbers above and below the sigma smbol. To do so, we need to solve the following equations: The first term of the series is 5: n + = 5 n = n = 5 n = n + The last term of the series is 7: n + = 7 n = 5 n = 5

Eample Write the following series in sigma notation: + 4 + 9 + 6 + 5 + 6 + 49 + 64 B inspection, we see that the series + 4 + 9 + 6 + 5 +6 + 49 + 64 is made up of the first eight perfect squares. 8 + 4 + 9 + 6 + 5 +6 + 49 + 64 = k Eample Write the following series in sigma notation: 6 + + 4 + 48 6 + + 4 + 48 is a geometric series with a = 6 and r =. Therefore: k = T n = a. r k = 6. k 6 = = k = k =. + k =. k... k =.... k The first term of the series is 6: so. k = 6 k = Divide both sides of the equation b k = 4 k =. k The last term of the series is 48: so. k = 48 k = 6 k = 4 k = 4 Eercise.6. Evaluate the following: 0 5 a) r b) (r + 5) c) r = 6 ( i r = 0 8 n = 6 0 ( 5 n ) d) e) i = ) f). k g) r k = r =. Write the following in sigma notation: a) + + 5 + + 7 b) + 9 + 7 + + 79 c) 7 + 0 + + + 5 d) 64 + + 6 + + e) 6 + 5 + 6 + + 00 f) 4 + 7 + 0 + + 7 g) + + 4 + + 0 h) + 4 + 6 + + 6 84 n = n 4

Series When we add the terms of an arithmetic progression we obtain the arithmetic series (S n ). When we add the terms of a geometric progression we obtain the geometric series (S n ). Eample If T n = n +, determine S 5. T T T T 4 T 5 S 5 = 5 + 8 + + 4 + 7 = 55 Also note: T T T T 4 T 5 5 + 8 + + 4 + 7 S 4 S 5 So: T 5 = S 5 S 4 = (5 + 8 + + 4 + 7) (5 + 8 + + 4) = 7 In general: T n = S n S n Eample Given that S n = n +, determine. the fifth term. T 5. T 5 = S 5 S 4 T 5 = [(5) + ] [(4) + ] T 5 = 5 5 T 5 = 8. T 5 = S 5 S 4 T 5 = [(5) + ] [(4) + ] T 5 = 5 55 T 5 = 98 The sum of an arithmetic series Find the sum of the arithmetic series + + + 4 + 5 + 6 + 7 + 8 + 9 + 0. We can write this as: S 0 = + + + 4 + 5 + 6 + 7 + 8 + 9 + 0 S 0 = 0 + 9 + 8 + 7 + 6 + 5 + 4 + + + Write the series backwards S 0 = + + + + + + + + + Add S 0 = 0 There are 0 terms S 0 = 0 0 S 0 = = 55 5

It can become quite tedious to add a large number of terms. Also, the more terms we have to add, the more chance there is of making a mistake. Fortunatel, we can use the same technique we have just used to develop a formula to add all the terms in a general arithmetic series. S n = T + T + T + + T n + T n = (a) + (a + d) + (a + d) + + (a + (n )d) + (a + (n )d) Proof of the formula for the sum of an arithmetic series: S n = a + ( a + d ) + ( a + d ) + + ( a + (n )d ) + ( a + (n )d ) + ( a + (n )d ) S n = (a + (n )d) + (a + (n )d) + (a + (n )d) + (a + d) + (a + d) + a S n = (a + (n )d) + (a + (n )d) + (a + (n )d) + + (a + (n )d) + (a + (n )d) S n = n(a + (n )d) Since there are n terms n(a + (n )d) S n = = n (a + (n )d) We can write the last term ( l )of an arithmetic series as l = a + (n )d. This means we can write the formula S n = n ( a + (n )d ) as S n = n ( a + a + (n )d ) Or S n = n ( a + l ) The sum of a geometric series Find the sum of the geometric series + + 4 + 8 + 6 + + 64 + 8. Here, a = ; r = and the number of terms is n = 8. The technique we use here is to multipl each term b the common ratio r. Therefore, we have: S 8 = + + 4 + 8 + 6 + + 64 + 8 S 8 = + 4 + 8 + 6 + + 64 + 8 + 56 Multipl b r = S 8 S 8 = + 0 + 0 + 0 + 0 + 0 + 0 + 0 56 ( )S 8 = 56 S 8 = 55 55 S 8 = = 55 Again, using the general form of a geometric series, we can determine a formula for the sum of n terms of a geometric series. S n = T + T + T + + T n + T n = a + ar + ar + + ar n + ar n 6

Proof of the formula for the sum of a geometric series: S n = a + ar + ar + + ar n + ar n r S n = ar + ar + + ar n + ar n + ar n S n rs n = a + 0 + 0 + + 0 + 0 ar n S n rs n = a ar n S n ( r) = a( r n ) Factorise S n( r) ( r) = a( r n ) ( r) a( r S n = n ) ( r) r In our proof, we subtracted equation from equation. This formula is easier to use if r <. We could also have subtracted equation from equation. This would give us a a(r slightl different formula: S n = n ). This formula is easier to use if r >. (r ) The proofs for both an arithmetic series formula and a geometric series formula must be known for eam purposes. Arithmetic series For an arithmetic series, we use the following formulae: S n = n (a + (n )d ) and S n = n (a + l ) Eample Determine the sum of the series + 5 + 8 + to 0 terms. a = ; d = ; n = 0; S 0 =? S n = n ( a + (n )d ) 0 S 0 = ( () + 9() ) = 0(6) = 60 Geometric series For a geometric series, we use the following formulae: a( r S n = n ) ( r) if r < a(r S n = n ) (r ) if r > Eample Use a formula to determine the sum of the series 7 + 9 + + to four terms. a = 7; r = _ ; n = 4; S =? 4 a( r S n = n ) r 7( ( _ S 4 = 4 ) _ 7( 8 S 4 = ) _ 8 S 4 = 7( 8 ) 7 S 4 = 80 8 S 4 = 40 7

Eample Determine + 7 + + + 59. a = ; d = 4; n =?; T n = 59; S n =? We first need to calculate the number of terms (n) in this series before we can calculate the sum. We use the formula T n = a + (n )d to calculate the number of terms. We also know the last term (l) in the series, so we can substitute n into the formula S n = n (a + l) to calculate the sum. T n = a + (n )d S n = n_ ( a + l ) 5 59 = + (n )4 S 5 = ( + 59 ) 5 59 = + 4n 4 S 5 = ( 6 ) 60 = 4n S 5 = 465 n = 5 Eample How man terms of the series + 4 + 7 + must be added to give a sum of 45? Eample Determine 8 + 4 + + + a = 8; r = _ ; n =?; T = n ; S =? n We first need to calculate the number of terms (n) in this series before we can calculate the sum. We use the formula T n = ar n to calculate the number of terms. Then, we substitute n into the formula a( r S n = n ) (r < ) to calculate the sum. r T n = ar n = 8 ( _ ) n _ = _ 8 8 8 ( _ ) n = 56 ( _ ) n ( _ ) 8 = ( _ ) n 8 = n 9 = n 8( ( _ ) 9 ) _ 8( 5) 5 5 ) _ S 9 = S 9 = S 9 = 8( S 9 = 8_ 5 ( 5) _ 5 S 9 = Eample How man terms of the series 5 + 0 + 0 + must be added to give a sum of 75? 8

a = ; d = ; n =?; S n = 45 S n = n_ ( a + (n )d ) 45 = n_ ( () + (n ) ) Multipl both 90 = n( + n ) sides b 90 = n(n ) 0 = n n 90 0 = (n + 9)(n 0) Factorise n = 9 or n = 0 Solve for n Since the number of terms is alwas a 9 natural number, n = is not valid. In other words, n cannot be fraction or a negative number. Therefore, n = 0. Eample 4 5 Determine (n 5) 5 n = n = (n 5) = + + 4 + + 70 a = ; d = ; n = 5; S 5 =? S n = n_ ( a + (n )d ) 5 S 5 = ( ( ) + 4() ) 5 S 5 = ( 68 ) S 5 = 850 a = 5; r = ; n =?; S n = 75 a( r S n = n ) r 5( 75 = n ) 75 = 5( n ) 55 = ( n ) 56 = n 8 = n n = 8 Eample 4 0 Evaluate (. n ) n = 0 (. n ) =. +. +. + n = +. 9 = 6 + + + 56 a = 6; r = ; n = 0 + = 9; S 9 =? a( r S n = n ) r Since r > 6(9 ) S 9 = S 9 = 6(5 ) = 066 Eercise.7. Calculate the sum of the following: a) + 7 + + to 5 terms b) 5 0 5 to 7 terms c) + 8 + + to 0 terms d) + 6 + to terms e) 5 9 to terms f) + 6 + 8 + to 0 terms. Determine the sum of the following series: a) 5 + + 9 + + 54 b) + 6 7 + 8 49 + + ( _ 7) 8 c) 75 70 65 + 0 d) 7 + 0 + + + 8 e) 9 + + + + f) 0 + 7 + 4 + 7 4 g) 4 + 0 + 4 + + 9 9

. Calculate the following: 0 5 0 a) r b) (r + 5) c) ) r 0 d) (9 n) r = r = r = 5 8 6 e). k f) ) n 8 g) (k ) k = n = 0 ( 4. How man terms of the following series must be added to give the indicated sum? a) + 7 + + = 0 b) + + 48 + = 4 095 c) 6 + + _ + = 765 d) 7 + 4 + + = 4 64 e) 7 5 = 5. Determine the sum of the first 50 even numbers. k = ( n = Mied problems Arithmetic sequences and series Geometric sequences and series Eample In an arithmetic sequence, T = 5 and T 6 =. Determine the sum of the first 0 terms of the sequence. T = 5 and T 6 = a + d = 5 a + 5d = a + 5d = (a + d) = 5 4d = 6 d = 4 a + 4 = 5 Substitute d = 4 into a = a = ; d = 4; n = 0; S 0 =? S n = n ( a + (n )d ) 0 S 0 = ( () + (0 )(4) ) S 0 = 0( + 9 4 ) S 0 = 780 Eample In a geometric sequence, T = 4 and T 6 = _. Determine the sum of the first 4 terms of the sequence if r > 0. T = 4 and T 6 = 4 ar = 4 ar 5 = _ 4 ar 5 ar = _ 4 4 r 4 = _ _ 4 4 r 4 = 6 r = _ since r > 0 a( _ ) = 4 Substitute r = into equation a = 8 a = 8; r = _ ; n = ; S =? a( r S n = n ) ( r) if r < 8 ( ( ) ) ( _ ) 8 ( 4 096 ) = 4 095 56 S = S = 0

Eample Determine the largest value of n such that n r = (r + 5) < 50 n r = (r + 5) = 7 + 9 + + + (n + 5) This is an arithmetic series with a = 7 and d =. S n = n ( (7) + (n ) ) = n ( n + ) = n + 6n n + 6n < 50 n + 6n 50 < 0 Since this trinomial does not factorise, we use the quadratic formula to solve for n. Solving the equation n + 6n 50 = 0, we get: b ± b n = 4ac a 6 4() ( 50) n = 6 ± () 6 ± 66 n = n = 5,6 or n = 9,6 If n + 6n 50 < 0 using a number line, we get: 5,6 9,6 0 0 5,6 < n < 9,6 the largest value of n is 9.

Eercise.8. In an arithmetic sequence, T = and T 7 = 6. Determine the sum of the first 0 terms of the sequence.. Determine the value of n if: n n a) (i 5) = 40 b). i = 85 i = n i = n c) (k + 7) = 008 d) (4 r) = 5 k = n r = n e) _ ( k ) = 640 f) _ ( k ) = 4 k =. In a geometric sequence, T = _ and T =. Determine the sum of the first 4 6 terms of the sequence. 4. In an arithmetic sequence, T 4 = and T 7 = 0. Determine the number of terms if the sum of the series is 60. Wh do ou obtain two values of n? 5. In an arithmetic sequence the seventh term eceeds the fourth term b 5. Determine: a) the value of d, the common difference. b) the value of a, if T 7 =. c) the tenth term. d) the sum of the first 5 terms. 6. In a geometric sequence, the fifth term is four times the third term, and the second term is 4. If r < 0, determine: a) the value of r, the common ratio. b) the value of a. c) the tenth term. d) the sum of the 5 terms. 7. a) Determine the largest value of n such that: n n i) (r + 4) < 0 ii) (i ) < 000 r = k = i = n b) What is the smallest value of n such that (5 k) < 550? k = 8. The first term of a geometric series is 9. The seventh term is 64. Determine two 8 possible values for the sum to seven terms of the sequence. 9. The sum of the first three terms of an arithmetic series is. The sith term is more than the fourth term. Determine: a) the common difference and the first term. b) the tenth term. 0. The sum of the first four terms of a geometric series is 7. The common ratio is. Calculate: 5 a) the first term. b) the seventh term.

. The numbers + ; 5 ; 7 + are the first three terms of an arithmetic sequence. Calculate: a) the value of. b) the sum of the first 0 terms of the sequence.. The numbers 4; + 8; + 0 are the first three terms of a geometric sequence. Calculate: a) the value of. b) the sum of the first si terms of the sequence. Applications of arithmetic and geometric sequences and series Arithmetic sequences and series Eample A ladder has rungs. The lowest rung is 800 mm long. Each succeeding rung is 40 mm shorter than the previous rung. Calculate the total length of rungs. The lowest rung is 800 mm The second rung will be 800 40 = 760 mm. The third rung will be 760 40 = 70 mm, and so on. So the sequence is: 800 + 760 + 70 + a = 800; d = 40; n = ; S =? S n = n (a + (n )d ) S = ((800) + ( 40)) S = 6( 60) S = 6 960 Geometric sequences and series Eample Michelle emails a letter to three of her friends. She asks them not to break the chain. The need to each forward the email to three other friends. If this process continues, determine how man people would have received the email if it is forwarded five times. Include the first time Michelle sent the e-mail. The sequence of the number of emails is: + 9 + 7 + a = ; r = ; n = 5 a(r S n = n ) r (5 ) S 5 = (4 ) S 5 = S 5 = 6

Eample A man s income is R96 000 a ear. Each ear, his income increases b R7 00. His epenses amount to R66 000 a ear, and increase b R4 00 ever ear. How long will it take him to save more than R80 940? Income: 96 000; 0 00; 0 400 Epenses: 66 000; 70 00; 74 400 Savings: 0 000; 000; 6 000 a = 0 000; d = 000; n =? S n > 80 940 S n = n_ ( a + (n )d ) n_ ( 60 000 + (n ) 000 ) > 80 940 n( 60 000 + 000n 000 ) > 6 880 000n + 57 000n 6 800 > 0 Using the quadratic formula: 57 000 ± 57 000 4( 000 ( 6 800 ) ) n = 000 n = 5,0 or n = 4,0 (N/A) So it will take si ears to save more than R80 940. Eample A ball is dropped from a height of m. The ball bounces back _ of the height of its previous bounce. Calculate the distance that the ball has travelled from the time it was dropped until it touches the ground for the fifth time. Round our answer to one decimal place. + ( ( _ ) + ( _ ) + ( _ ) + ( _ ) 4 ) a = ( _ ); r = _ ; n = 4 a( r S n = n ) r ( _ S 4 = )( ( _ ) 4 ) ( _ ) 8( ( 6 8 S 4 = ) ) _ 8 6 S 4 = 8( ) _ 8 S 4 = 8_ 65 _ 8 S 4 = 9,6 m the total distance travelled is: + (9,6) = 50,5 m Eercise.9. Farmer Langa starts farming with 450 sheep. He finds that his stock increases b % each ear. How man sheep will Langa have on his farm at the end of five ears?. Farmer Joe starts farming with a certain number of cattle. He finds that, each ear, he has 0 more cattle. At the end of five ears, the farmer has 40 cattle. How man cattle did the farmer start with?. Dean s grann gives him R on his first birthda, R on his second birthda, R on his third birthda, R4 on his fourth birthda, and so on. a) How much mone will Dean receive on his 0th birthda? b) Calculate the total amount of mone Dean would have received from his grann over the 0 ears. 4

4. Sipho s grann gives him one cent on his first birthda, two cents on his second birthda, four cents on his third birthda, eight cents on his fourth birthda, and so on. a) How much mone will Sipho receive on his 0th birthda? b) Calculate the total amount of mone Sipho would have received from his grann over the 0 ears. 5. Refer to questions and 4. Would Dean or Sipho have received more mone over the 0 ears? 6. An athlete is training to run the Comrades Marathon. He runs km on the first da and increases his distance b km each da. a) On which da would he cover a distance of km? b) After how man das would he have covered a total distance of 0 km? 7. The road works department is tarring a road. The set up camp at the start of the road. The workers manage to tar 0,6 km of road ever da and return to their camp site at the end of ever da. a) How far will the workers travel on the 0th da? b) How far will the workers have travelled in total after 0 das? 8. Kashiv saves R500 in the first month of his working career. He saves the same amount at the end of each month of the ear. Each subsequent ear, he manages to save 0% more than he saved the previous ear. Calculate: a) Kashiv s total savings at the end of the first ear b) the amount that Kashiv would be saving monthl in his sith ear c) the total amount that Kashiv would have saved at the end of si ears. 9. A horizontal line intersects part of the sine curve at four points. It therefore divides the curve into five parts. a) If a second line is drawn to intersect the curve, into how man parts will the curve be divided? b) If ten lines are drawn to intersect the curve, how man parts will the curve be divided into? 0. A factor manufactures a product for R00,00. Each time the product is bought and sold a profit of 5% is made. a) If the product is bought and sold seven times, what will the price of the product be? b) Calculate the difference between the original price and the price after it has been sold for the seventh time.. Vaughan is preparing for a biccle race. In the first week, he rides km. He then increases his distance b km each week. a) What distance did Vaughan ride in the seventh week? b) What was the total distance Vaughan covered after seven weeks? 5

. A water tank contains 6 l of water at the end of da. Because of a leak, the tank loses one-sith of the previous da s contents. How man litres of water will there be in the tank at the end of the: a) second da b) third da c) seventh da?. Zintle decides to join a stokvel to save for her son s education when he leaves school. She joins the stokvel in Januar of his Grade ear and has to pa R00,00 monthl. The stokvel paments increase b R50,00 each ear. If there are people in the stokvel and Zintle is paid in December, how much will she have saved at the end of ears, assuming that she does not spend an of her stokvel paments? Infinite series Eercise.0 For each series: a) Complete the table. b) Plot a graph where the -ais represents the number of terms and the -ais represents the sum of the terms.. + + 4 + 8 + S S S S 4 S 5 S 6 S 7 S 8 S 9 S 0 Graph for series : 000 900 800 700 600 500 400 00 00 00 6 0 4 5 6 7 8 9 0

. 5 + 5 45 S S S S 4 S 5 S 6 S 7 S 8 S 9 S 0 Graph for series : 75 000 60 000 45 000 0 000 5 000 0 5 000 4 5 6 7 8 9 0 0 000 45 000 60 000 75 000. + _ + _ + _ + 4 8 S S S S 4 S 5 S 6 S 7 S 8 S 9 S 0 Graph for series :,00,75,50,5,00 0,75 0,50 0,5 0 4 5 6 7 8 9 0 7

4. 6 4 + 8 6 9 + S S S S 4 S 5 S 6 S 7 S 8 S 9 S 0 Graph for series 4: 6 5 4 0 4 5 6 7 8 9 0 5. + 7 + + S S S S 4 S 5 S 6 S 7 S 8 S 9 S 0 Graph for series 5: 0 00 80 60 40 0 00 80 60 40 0 0 4 5 6 7 8 9 0 8

6. 6 + 4 S S S S 4 S 5 S 6 S 7 S 8 S 9 S 0 Graph for series 6: 0 0 0 4 5 6 7 8 9 0 40 60 80 00 0 40 60 80 7. 7 + 9 + + S S S S 4 S 5 S 6 S 7 S 8 S 9 S 0 Graph for series 7: 40 5 0 5 0 5 0 5 0 4 5 6 7 8 9 0 9

8. 6 + 9 + 7 + S S S S 4 S 5 S 6 S 7 S 8 S 9 S 0 Graph for series 8: 600 500 400 00 00 00 0 4 5 6 7 8 9 0 9. 8 + S S S S 4 S 5 S 6 S 7 S 8 S 9 S 0 Graph for series 9: 5 0 5 0 5 0 5 0 4 5 6 7 8 9 0 0

0. 5 5 5 S S S S 4 S 5 S 6 S 7 S 8 S 9 S 0 Graph for series 0: 0 5 4 5 6 7 8 9 0 0 5 0 5 0 5 40 45 50. 5 + 9 + 7 5 + S S S S 4 S 5 S 6 S 7 S 8 S 9 S 0 Graph for series : 40 5 0 5 0 5 0 5 0 4 5 6 7 8 9 0

. 6 8 + 4 S S S S 4 S 5 S 6 S 7 S 8 S 9 S 0 Graph for series : 6 4 0 8 6 4 0 4 5 6 7 8 9 0. 4 + _ + + S S S S 4 S 5 S 6 S 7 S 8 S 9 S 0 Graph for series : 8 6 4 0 8 6 4 0 4 5 6 7 8 9 0

4. 5 + 5 + 6 + S S S S 4 S 5 S 6 S 7 S 8 S 9 S 0 Graph for series 4: 80 70 60 50 40 0 0 0 0 4 5 6 7 8 9 0 Until now, we have onl ever added a finite number of terms. An infinite series has an infinite number of terms. In some cases, we can work out the sum of an infinite series. We write this as S, which means the sum to infinit. If ou could add an infinite number of terms, what do ou think would be the sum of the series? Using the tables and graphs from the previous questions, complete the following table: Sequence AP or GP d = r = Conclusion about the S 4 5 6 7 8 9 0

Sequence AP or GP d = r = Conclusion about the S 4 Tick the relevant block: A series: converges if the sum approaches a particular value as we add more terms. diverges if the sum of the series becomes a ver large positive or negative number as we add more terms. oscillates if, as we add more terms, the sum of the series changes between being positive and negative. Series Geometric series Arithmetic series r < < r < r = r > d < 0 d > 0 Converges Diverges Oscillates Conclusion: Sum to infinit From the investigation, we find that onl a geometric series will converge. In fact, a geometric series will onl converge if < r <. We can derive a formula for the sum to infinit as follows: a ar S n = n r a S n = r ar n r if < r <, then r n 0 as n (Remember that the arrow ar n 0 as n means tends to ) r a S n r as n a S = r 4

Eample Determine:. ( ) n =. ( ) n n = a = r = a S = r = = = = 4 n =. ( = + + + 4 + ) 0 +. ( ) +. ( ) +. ( ) + Eample Use the formula for S of a geometric series to epress 0, 6as a common fraction. 0, 6 6 = 0 + 6 00 + 6 000 + 6 a = 0 r = 0 a S = r 6 0 = 6 0 = 9 0 6 0 = 0 0 9 = Eample For which values of will the series ( + ) + ( + ) + ( + ) + converge? For the series ( + ) + ( + ) + ( + ) + r = + For the series to converge, < r < < + < < < 0 5

Eercise.. Determine the sum to infinit for the following series: a) 7 + 9 + + + b) 4; ; ; ; c) 6 4 + 4 + d) 6 + 4 + 8 + 6 9 + 7 e) + 6 8 + 4 f) 5 + 5 + 9 + 5 +. Evaluate the following, if possible. If not possible, give a reason wh the sum to infinit cannot be found. n n a). n = ( ) b) ( 4 n = 5) c) () n n = d) 8( ) n e) 8( ) n f) n = 0 n = n =. Convert each of the following to a common fraction: a) 0, 8 b), 4 c), 5 4. Given the sequence 5(4 5 ) + 5(4 4 ) + 5(4 ) + a) Show that the series is convergent. b) Calculate the sum to infinit of the series. 5. Given the geometric series 9 + + + a) Show that T n = 7( ) n b) For which values of will the series converge? c) Calculate the sum to infinit if =. 6. In a sequence of squares, the sides of the first square are 4 cm long. The sides of each subsequent square are half that of the previous square. 4 cm cm cm cm a) Determine the length of the side of the eighth square. b) Write down the series for the perimeter of the squares. c) Determine the sum of the perimeters of the squares if the continue infinitel. d) Write down the series for the areas of the squares. e) Determine the sum of the areas of the squares if the continue infinitel. 6

7. The numbers 5m 7; m + ; m + are positive numbers and the first three terms of a convergent geometric series. Calculate: a) the value of m. b) the sum to infinit of the series. 8. The sum to infinit of a geometric series is 8 and the common ratio is 4. Calculate the first term of the series. 9. In the series 6 + + + +, A is the sum to infinit and B is the sum to n 4 terms. Calculate: a) the value of A b) the value of B in terms of n c) the value of n for which is A B = 64. 0. A plant is a 00 cm tall when planted. At the end of the first ear, the plant is 0 cm tall. Each ear, the plant grows b half the amount of the previous ear. a) What will the height of the plant be after si ears? b) Show that the plant will never eceed a height of 40 cm. Summar Arithmetic Geometric Test T T = T T T T = T T General form T n = a + (n )d T n = a. r n Sum Mean S n = n ( a + (n )d ) or S n = n ( a + l ) a( S n = r n ) ; r < r or a( S n = r n ) r ; r > a + b ab Converges for: < r < a S S = r 7

Revision eercise. Given the following sequence: ; 6; ; a) Show that T n =. n 0 b) Determine: T n n =. Given the series: 6; ; 9; 7 4 ; a) Calculate the sum of the first ten terms of the series. b) Determine the sum to infinit. c) Write the sum of the first ten terms of the series in sigma notation. n n(n + ). a) Prove that (r + 4) = r = b) Hence find the sum of the first 0 terms of the series. c) How man terms of the series will give a sum of 996? 4. How man terms of the series + 5 + 8 + + add up to 876? 5. In an arithmetic progression, S 6 = 0 and S 5 =. Determine the value of T 6 6. For which values of will the geometric series + ( + ) + ( + ) + be convergent? 7. The sum of the third and the seventh terms of an arithmetic series is 48. The sum of the first ten terms of the series is 65. Determine the first three terms of the series. 8. T T T T 4 T 5 4 8 This sequence could be either arithmetic or geometric. a) Determine the nth term in each case. b) Hence, determine the first and fourth terms if the sequence is: i) arithmetic ii) geometric. 9. Given the sequence: ; ; 4 ; 5; ; 9; 8 a) Write down the net four terms if the sequence continues in the same manner. b) Determine the sum of the first 40 terms of the sequence. 0. Determine the nth of the sequence 4; ; if the sequence is a) arithmetic b) geometric. 8

. The sum of the first n terms of a series is given b the formula S n = n + 9. a) Determine the sum of the first 0 terms of the series. b) Determine the 0th term of the series. c) Show that T n =. n.. Determine the 5th term of the arithmetic progression: + ; ; ;. A ball drops from a height of 6 metres and rebounds half its distance on each bounce. Calculate the total distance it will have travelled before coming to rest. 4. The sum of n terms of the arithmetic series + 5 + 8 is equal to the sum of n terms of the arithmetic series 5 + 6 + 8 + 9 + Calculate the value of n. 5. A tree is planted and the height is measured at the end of each ear. The height of the tree is found to be m at the end of the first ear. In the second ear, the tree increases in height b 5 cm. The tree increases in height each ear b 4 of the 5 previous ear s growth. a) Complete the table: Year Year Year Year 4 Year 5 Height of tree in m,5 Growth in cm 5 b) Determine the increase in the height of the tree at the end of the th ear. c) Determine the height of the tree after ears. d) Show that the maimum height the tree will reach will be,75 m. 6. Since 00, the deaths per 00 000 people at risk due to malaria in Africa have roughl followed the following pattern: T T T T 4 T 5 00 004 005 006 007 6,, 9,6 07 a) Determine whether this follows the pattern of i) a geometric sequence ii) an arithmetic sequence iii) a quadratic sequence. b) Determine the nth term of this sequence. c) What percentage reduction has taken place between 00 and 00? 9

Chapter Functions and inverse functions In this chapter ou will: revise functions dealt with in grades 0 and define a function learn about the inverse of a function learn how to sketch the graphs of inverse functions. In grades 0 and, ou learnt how to draw the graphs of different functions, namel: the straight line: = a + q the parabola: = a( + p) + q a the hperbola: = ( + p) + q the eponential function: = a. b + p + q Revising functions dealt with in Grade 0 and Before sketching a graph: ou need to identif the curve, so ou need to be familiar with the general equation of each function draw a rough sketch of the graph calculate the possible - and -intercepts of the graph write down the equations of an asmptotes write down the equation of the ais of smmetr of a parabola and the coordinates of the turning point. 40

Eample Sketch the graph of = + This is the graph of a straight line with a negative gradient: gradient = Rough sketch: To calculate the -intercept, we let = 0 and solve for : = ( 0 ) + = The -intercept is (0; ). To calculate the -intercept, we let = 0 and solve for : 0 = + 0 = + 6 Multipl both sides b = 6 The -intercept is (6; 0). 6 5 4 0 4 5 6 7 4 4

Eample Sketch the graph of = ( ) 8 This is the graph of a parabola. The equation of the ais of smmetr is =. The turning point of the graph is (; 8). To calculate the -intercept, let = 0 and solve for : = (0 ) 8 = 0 The -intercept is (0; 0). Rough sketch: To calculate the -intercept, let = 0 and solve for : 0 = ( ) 8 8 = ( ) Add 8 to both sides 4 = ( ) Divide both sides b ± 4 = ( ) Take the square root of both sides = or = = = 5 0 The -intercepts are (; 0) and (5; 0). 9 8 7 6 5 4 0 4 5 6 7 8 4 5 6 ( ) 8 4

Eample Sketch the graph of = 4 +. This is the graph of a hperbola. The equation of the asmptotes are = and =. To calculate the -intercept, we let = 0 and solve for : = 4 0 + = 6 The -intercept is (0; 6). To calculate the -intercept, we let = 0 and solve for : 0 = 4 + 4 = Add 4 to both sides 4 = ( ) Find the LCD 4 = Multipl out the bracket 6 = Add to both sides = Divide both sides b The -intercept is (; 0). Determine the coordinates of other points to help ou draw the graph. 9 8 7 6 5 4 Rough sketch: 0 4 5 6 7 4 5 6 7 4 = ( ) 4

Eample 4 Sketch the graph of = ( ) + + 4 This is the graph of an eponential Rough sketch: function. The equation of the asmptote is = 4. To calculate the -intercept, we let = 0 and solve for : = ( ) 0 + + 4 = The -intercept is (0; ). To calculate the -intercept, we let = 0 and solve for : 0 = ( ) + + 4 ( ) + = 4 Add ( ) + t o both sides ( ) + = Divide both sides b ( ) + = Write as = Raising a power to a power = = = The -intercept is ( ; 0). Equate the eponents Solve for 4 = + + 4 0 44

Eercise.. Sketch the graphs of the following: a) = b) = ( ) c) = d) = e) = f) = g) = h) = i) = + + j) = ( + ) + k) = l) = m) = n) = o) = ( ) + 4 p) = + q) = ( ) + 4 r) = + s) = ( ) 8 t) =. +. Determine the equations of the following: a) f() = a( + p) + q b) g() = (; 9) a + p (0; 5) f (; ) c) h() = ab + q d) p() = a + b + c (; 7) ( ; 6) 45

a e) g() = + p + q f) h() =. b + p + q 8 6 ( 4; ) ( ; ) g) = a + p ; > 0 h) = b + p (; ) i) = a ( ; ) 46

Functions A relation is an relationship between two variables. A function is a special kind of relation in which: For ever -value, there is at most one -value. Each element of the domain is associated with onl one element of the range. In other words, the -values are never repeated in the set of ordered pairs of a function. For eample: {(; ); (; 4); (; 6)} is a function {(; ); (; ); (; 4); (; 4)} is a NOT a function, because the -coordinates are repeated. An vertical line will cut the graph of a function once and onl once. For eample: Function Not a function Not a function A function has one-to-one or man-to-one mapping. i) ii) iii) iv) 4 4 4 4 5 5 5 5 0 6 0 6 0 6 0 6 7 7 7 7 8 8 8 8 9 9 9 9 One-to-one mapping function Man-to-one mapping function One-to-man mapping not a function Man-to-man mapping not a function If we list the ordered pairs for each mapping, we have: i) {( ; 5); (0; 7); (; 9)} No -coordinate is repeated, so the relation is a function. Ever -value maps onto onl one -value. In other words, neither the - nor the values are repeated. The domain is { ; 0; } and the range is {5; 7; 9}. ii) {( ; 5); ( ; 5); (0; 7); (; 7)} No -coordinate is repeated, so the relation is a function. Man -values map onto more than one -value. In other words, the -values are not repeated, but the -values are repeated. The domain is { ; ; 0; } and the range is {5; 7}. 47

iii) {( ; 4); ( ; 4); ( ; 6); (; 7); (; 8)} Two -coordinates are repeated, so the relation is not a function. The same -value maps onto different -values. In other words, the -values are repeated, but the -values are not repeated. The domain is { ; ; } and the range is {4; 6; 7; 8}. iv) {( ; 4); ( ; 4); ( ; 6); (; 6); (; 8)} Two - coordinates are repeated, so the relation is not a function. Man -values map onto man -values. In other words, both the - and the values are repeated. The domain is { ; ; } and the range is {4; 6; 8}. A function is increasing if the variables change in the same direction. In other words, as the -values increase, the -values increase as well. Or, as the -values decrease, the -values decrease as well. A function is decreasing if the variables change in different directions. In other words, as the values of increase, the values of decrease. Or, as the values of decrease, the values of increase. Eercise.. Determine which of the following graphs are functions. a) b) c) d) e) f) g) h). f = {(; 5); (; 7); (4; 9); (5; )} a) Is f a function? Give a reason for our answer. b) Write down the domain and range of f. c) Determine m if f(m) = 9. d) Determine n if f() = n.. Given that P is not a function, determine the value(s) of. P = {( ; ); ( + ; ); (4; 5)} 48

4. Do the following mappings represent a function? Give a reason for our answer. a) 0 b) 4 4 6 8 0 0 4 c) 5 0 7 9 4 5 6 7 8 9 d) 4 5 6 4 0 9 5. Write down the domain and range for each of the mappings in question 4. Inverse functions The inverse of a one-to-one function Operation add subtract multipl divide square square root Inverse operation subtract add divide multipl square root square Eample Find the inverse of the function f() = + 5 If we draw a flow diagram of this function, it will look as follows: + 5 Multipl b Add 5 Divide b + 5 If we now replace each operation with its inverse, the flow diagram will look as follows: 5 5 divide b subtract 5 multipl b 49