SAMPLE CHAPTER INTERNATIONAL GCSE (9 1) ALAN SMITH SOPHIE GOLDIE. Mathematics. for Edexcel Specification A THIRD EDITION

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1 SAMPLE CHAPTER INTERNATIONAL GCSE (9 1) ALAN SMITH SOPHIE GOLDIE Mathematics for Edexcel Specification A THIRD EDITION 1

2 Ensure complete coverage of the latest Edexcel (9-1) International GCSE Mathematics A Specification, with our fully updated and highly popular Student Book and Practice Book, written by experienced examiners, teachers and authors. The following titles will be entered in to Edexcel s endorsement process: Edexcel International GCSE (9-1) Mathematics Student Book Third Edition September Edexcel International GCSE (9-1) Mathematics Practice Book Third Edition September Visit to pre-order your class sets or to sign up for your Inspection Copies or einspection Copies. Our Student Book will also be available in two digital formats via Dynamic Learning: Student etextbooks provide a downloadable version of the printed textbook that you can assign to students so they can: Download and view on any device or browser Add, edit and synchronise notes across any two devices Complete interactive, self-check questions 6.99 per student for 1 year s access / per student for 2 year s access / per student for 3 year s access Publishing from October 2016 Whiteboard etextbooks are online, interactive versions of the printed textbooks that enable you to: Display pages for whole-class teaching Add notes and highlight areas Insert double-page spreads into your lesson plans and homework activities 200 (small institution up to 900 students) 250 (large institution 901+ students) Publishing from Sept 2016 To find out more about etextbooks visit: CPD training Ensure that you are fully prepared for the upcoming changes by attending one of our new specification courses. For more information and to book your place visit

3 INTERNATIONAL GCSE (9 1) ALAN SMITH SOPHIE GOLDIE Mathematics for Edexcel Specification A THIRD EDITION

4 Contents 1 Fractions, decimals and rounding Starter: Half and half 1.1 Equivalent fractions 1.2 Multiplying and dividing with fractions 1.3 Decimals and fractions 1.4 Rounding and approximation 1.5 Rounding calculator answers 1.6 Upper and lower bounds Internet Challenge 1 2 Ratios and percentages Starter: How many per cent? 2.1 Working with ratios 2.2 Simple percentages 2.3 Percentage increase and decrease 2.4 Reverse percentage problems 2.5 Compound interest Internet Challenge 2 3 Powers and roots Starter: Roman numerals 3.1 Basic powers and roots 3.2 Higher powers and roots 3.3 Fractional (rational) indices 3.4 Negative powers 3.5 The laws of indices 3.6 Standard index form 3.7 Calculating with numbers in standard form 3.8 Factors, multiples and primes 3.9 Highest common factor, HCF 3.10 Lowest common multiple Internet Challenge 3 4 Working with algebra Starter: Right or wrong? 4.1 Substituting numbers into formulae and expressions 4.2 Working with indices 4.3 Expanding brackets 4.4 Multiplying two brackets together 4.5 Factorising common factors 4.6 Factorising quadratic expressions 4.7 Factorising harder quadratic expressions 4.8 Factorising difference of two squares 4.9 Generating formulae 4.10 Changing the subject of a formula Internet Challenge 4 5 Algebraic equations Starter: Triangular arithmagons 5.1 Expressions, equations and identities 5.2 Simple equations 5.3 Harder linear equations 5.4 Equations and brackets 5.5 Equations with fractional coefficients Internet Challenge 5 6 Graphs of straight lines Starter: Matchstick puzzles 6.1 Coordinates in all four quadrants 6.2 Graphs of linear functions 6.3 Gradient and intercept of linear functions 6.4 Equations and graphs 6.5 Parallel and perpendicular lines Internet Challenge 6 7 Simultaneous equations Starter: Fruity numbers 7.1 Solving simultaneous equations by inspection 7.2 Solving simultaneous equations by algebraic elimination 7.3 Solving simultaneous equations by a graphical method 7.4 Setting up and solving problems using simultaneous equations 4

5 Contents Internet Challenge 7 8 Inequalities Starter: Treasure hunt 8.1 Whole-number solutions to inequalities 8.2 Using algebra to solve linear inequalities 8.3 Illustrating inequalities on a number line 8.4 Graphs of linear inequalities in two variables 8.5 Quadratic inequalities Internet Challenge 8 9 S Number sequences and seies Starter: Circles, lines and regions 9.1 Number sequences 9.2 Describing number sequences with rules 9.3 Arithmetic sequences 9.4 Arithmetic series Internet Challenge 9 10 Travel and other graphs Starter: Animal races 10.1 Distance time graphs 10.2 Modelling with graphs Internet Challenge Working with shape and space Starter: Alphabet soup 11.1 Corresponding and alternate angles 11.2 Angles in triangles and quadrilaterals 11.3 Angles in polygons 11.4 Areas and perimeters of simple shapes 11.5 Surface area and volume Internet Challenge Circles, cylinders, cones and spheres Starter: Three and a bit 12.1 Circumference and area of a circle 12.2 Sectors of a circle 12.3 Circumference and area in reverse 12.4 Surface area and volume of a cylinder 12.5 Exact calculations using pi 12.6 Volume and surface area of cones and spheres Internet Challenge Geometric constructions Starter: Round and round in circles 13.1 Constructing triangles from given information 13.2 Constructions with line segments 13.3 Bearings Internet Challenge Transformation and similarity Starter: Monkey business 14.1 Reflections 14.2 Rotations 14.3 Combining transformations 14.4 Enlargements 14.5 Similar shapes and solids Internet Challenge Pythagoras theorem Starter: Finding squares and square roots on your calculator 15.1 Introducing Pythagoras theorem 15.2 Using Pythagoras theorem to find a hypotenuse 15.3 Using Pythagoras theorem to find one of the shorter sides 15.4 Pythagoras theorem in three dimensions 15.5 Pythagoras theorem on a coordinate grid Internet Challenge Introducing trigonometry Starter: A triangular spiral 16.1 The sine ratio 16.2 The cosine ratio 5

6 Contents 16.3 The tangent ratio 16.4 Choosing the right trigonometrical function 16.5 Finding an unknown angle 16.6 Multi-stage problems 16.7 Angles of elevation and depression Internet Challenge Circle theorems Starter: Circle vocabulary 17.1 Tangents, chords and circles 17.2 Angle properties inside a circle 17.3 Further circle theorems 19.4 Intersecting chords Internet Challenge Sets Starter: Does it all add up? 18.1 Introducing set notation 18.2 Venn diagrams 18.3 Further Venn diagrams Internet Challenge Working with data Starter: Lies, damned lies and statistics 19.1 Calculations with frequency tables 19.2 Solving problems involving the mean 19.3 Histograms 19.4 Cumulative frequency 19.5 Median and quartiles for a discrete data set Internet Challenge Probability Starter: Dice throws 20.1 Theoretical and experimental probability 20.2 Mutually exclusive outcomes 20.3 Independent events 20.4 Tree diagrams Internet Challenge Direct and inverse proportion Starter: A sense of proportion 21.1 Direct proportion 21.2 Inverse proportion 21.3 Graphical representation of direct and inverse proportion 21.4 Compound measures Internet Challenge Quadratic equations, curves and inequalities Starter: Solutions of equations 22.1 Solving quadratic equations factorising 22.2 Completing the square 22.3 Solving quadratic equations formula 22.4 Problems leading to quadratic equations 22.5 Quadratic curves 22.6 Solving inequalities Internet Challenge Advanced algebra Starter: How many shapes? 23.1 Working with surds 23.2 Algebraic fractions 23.3 Cancelling common factors in rational expressions 23.4 Simultaneous equations, one linear and one quadratic 23.5 Changing the subject of an equation where the symbol occurs twice 23.6 Algebraic proofs Internet Challenge Functions and function notation Starter: Number crunchers 24.1 Introducing functions and function notation 24.2 Domain and range 24.3 Inverse functions 24.4 Composite functions Internet Challenge 24 6

7 Contents 25 Further trigonometry Starter: How tall is the church? 25.1 The sine rule 25.2 The cosine rule 25.3 Area of a triangle using } 1 2 } ab sin C, and segments of circles 25.4 Trigonometry in 3-D Internet Challenge Graphs and transformations Starter: Making waves 26.1 Plotting and using graphs of curves 26.2 Graphs of sine, cosine and tangent functions Internet Challenge Vectors Starter: Knight s tours 27.1 Introducing vectors 27.2 Adding and subtracting vectors 27.3 Multiplying a vector by a number (scalar multiplication) 27.4 Using vectors Internet Challenge Calculus Starter: Steeper and steeper 28.1 Gradient of a curve 28.2 Gradient of a curve differentiation 28.3 Harder differentiation 28.4 Maximum and minimum points on curves 28.5 Further problems on maximum and minimum 28.6 Distance, velocity and acceleration Internet Challenge 28 7

8 Chapter 9: Number sequences CHAPTER 9 Number sequences In this chapter you will learn how to: recognise and use common number sequences use rules to generate number sequences find a general formula for the nth term of an arithmetic sequence find the sum of an arithmetic series. You will also be challenged to: investigate Fibonacci numbers. Look at the sequence of circles below. 1 1 Starter: Circles, lines and regions Pattern 1 1 point 0 lines 1 region Pattern 2 2 points 1 line 2 regions 2 4 Pattern 3 3 points 3 lines 4 regions The diagram shows a sequence of circles. Each circle has some points marked around its circumference. Each point is joined to every other point by a line. The lines and regions are then counted. The lines and regions are not all the same size. Task 1 Describe a rule for how the number of points increases in this sequence. Task 2 Describe a rule for how the number of lines increases. Task 3 Describe a rule for how the number of regions increases Pattern 4 4 points 6 lines 8 regions Highlighted content is new for this edition. Each chapter begins with a STARTER: a problem-solving exercise, activity or puzzle designed to stimulate thinking and discussion about some of the ideas that underpin the content of the chapter. 8

9 9.1 Number sequences Task 4 Now draw pattern 5 and pattern 6, and see if your rules seem correct. You should space out the points so that no triple intersections can occur, otherwise you lose a region, for example: No Yes 9.1 Number sequences Here are some number sequences that occur often in mathematics. Name of sequence First six terms Formula for the nth term Positive integers 1, 2, 3, 4, 5, 6, n Even numbers 2, 4, 6, 8, 10, 12, 2n Odd numbers 1, 3, 5, 7, 9, 11, 2n 1 Square numbers 1, 4, 9, 16, 25, 36, n 2 Cube numbers 1, 8, 27, 64, 125, 216, n 3 Powers of 2 2, 4, 8, 16, 32, 64 2 n Powers of 10 10, 100, 1000, , , 10 n , You may encounter these number patterns when solving mathematical problems based on counting patterns. EXAMPLE Look at this pattern of squares. Pattern 1 Pattern 2 Pattern 3 Pattern 4 a) How many squares would there be in pattern 5? b) Find a formula for the number of squares in pattern n. c) Use your formula to find the number of squares in pattern

10 Chapter 9: Number sequences SOLUTION The number of squares forms a pattern 2, 4, 6, 8, that is, the even numbers. a) Pattern 5 contains squares. b) Pattern n contains 2n squares. c) Pattern 100 contains squares. Some number sequences are disguised versions of the common ones, perhaps with a constant number added or multiplied. EXAMPLE Find the next three terms in this number sequence. Find also a formula for the nth term. 101, 104, 109, 116, 125, SOLUTION 101, 104, 109, 116, 125, are all 100 more than the square numbers. The next three terms are , and , that is, 136, 149, 164 The nth term is n 2 EXERCISE 9.1 Write down the next two terms in each of these number sequences, and explain how each term is worked out. Give an expression for the nth term in each case. Questions to test students understanding on each sub-topic. They are all related to the list of common sequences in the table on the previous page. 1 10, 20, 30, 40, 50, 60, 2 5, 7, 9, 11, 13, 15, 3 51, 53, 55, 57, 59, 61, 4 4, 8, 12, 16, 20, 24, 5 2, 8, 26, 80, 242, 6 0.1, 0.01, 0.001, , 7 10, 30, 60, 100, 150, 210, 8 2, 8, 18, 32, 50, 72, 9 Look at this pattern of triangles. Pattern 1 Pattern 2 Pattern 3 Pattern 4 a) How many triangles would there be in pattern 7? b) Find a formula for the number of triangles in pattern n. 10

11 9.2 Describing number sequences with rules 10 Look at this pattern of spots. Pattern 1 Pattern 2 Pattern 3 Pattern 4 a) Find an expression for the number of spots in pattern n. b) How many spots would there be in pattern 30? 9.2 Describing number sequences with rules It can be very useful to be able to describe number sequences using rules. One way of doing this is to say how each term is connected to the next one in the sequence. (This is sometimes called a term-to-term rule, because it explains the link between one term and the next.) Each sub-topic within each chapter is introduced, and explained through easy-to-follow worked examples with solutions. EXAMPLE A number sequence is defined as follows: The first term is 3. Each new term is double the previous one. Use this rule to generate the first five terms of the number sequence. SOLUTION Start with 3: etc. The first five terms of the sequence are 3, 6, 12, 24, 48,. EXAMPLE A number sequence is defined as follows: The first term is 7. Each new term is 3 more than the previous one. Use this rule to generate the first six terms of the number sequence. 11

12 Chapter 9: Number sequences SOLUTION Start with 7: etc. The first six terms of the sequence are 7, 10, 13, 16, 19, 22,. If you wanted to work out the 100th number in a sequence, it would be very tedious to have to write out all 100 numbers, one at a time. In this case it is better if you can use an algebraic expression for the nth term. (This is sometimes called a position-to-term rule, since you can work out the value of any term as long as you know its position in the sequence.) EXAMPLE The nth term of a number sequence is given by the expression 2n a) Write down the first four terms of the sequence. b) Find the value of the 20th term. SOLUTION a) n 5 1 gives n 5 2 gives n 5 3 gives n 5 4 gives The first four terms are 3, 9, 19, 33 b) When n 5 20, EXERCISE A number sequence is defined as follows: The first term is 5. Each new term is 2 more than the previous one. Use this rule to generate the first five terms of the number sequence. 2 A number sequence is defined as follows: The first term is 1. To find each new term, add 1 to the previous term, and double this total. Use this rule to generate the first four terms of the number sequence. 3 The nth term of a number sequence is given by the expression 8n 2 1. a) Write down the values of the first five terms. b) Work out the value of the 20th term. 12

13 9.3 Arithmetic sequences 4 The nth term of a number sequence is given by the expression 3n a) Write down the values of the first six terms. b) Work out the value of the 23rd term. 5 Andy has been doing a mathematical investigation. He gets this sequence of numbers: 12, 15, 18, 21, 24, a) Describe Andy s pattern in words. b) Find the tenth term in Andy s number sequence. 6 The nth term of a number sequence is given by the expression n. a) Write down the values of the first five terms. b) Work out the value of the 50th term. 7 In a certain number sequence, the first term is 3. Each new term is found by multiplying the previous term by 3. a) Write down the first five terms of the number sequence. b) What name is given to this particular number sequence? 8 The nth term of a number sequence is given by the formula 7n 1 3. a) Work out the first three terms. b) Find the value of the 10th term. c) One of the numbers in the sequence is Which term is this? 9 The nth term of a number sequence is given by the expression n(n 1 1). 2 a) Write down the values of the first four terms. b) Work out the value of the 30th term. c) Explain why all the terms in this sequence are integers. d) What name is often given to the number sequence generated by this rule? Problem-solving question. 10 David is working with a number sequence. The nth term of his sequence is given by the expression 6n 1 7. He gets the number 2770 as one of his terms. Show that David must have made a mistake. 9.3 Arithmetic sequences A number sequence in which the terms go up or down in equal steps is called an arithmetic sequence. The size of the step is called the common difference. The first term of an arithmetic sequence is a. The common difference is d. EXAMPLE For each sequence, say whether it is arithmetic or not. For each arithmetic sequence state the value of the first term, a, and the common difference, d. a) 2, 3, 5, 8, 12, b) 2, 5, 8, 11, 14, c) 1, 2, 4, 8, 16, d) 40, 36, 32, 28, 24, When d is negative, each term is less than the preceding term. Higher objective 3.1 A is covered understand and use common difference (d) and first term (a) in an arithmetic sequence. Callout boxes add clarity: new for this edition. 13

14 Chapter 9: Number sequences SOLUTION a) 2, 3, 5, 8, 12, is not an arithmetic sequence. b) 2, 5, 8, 11, 14, is an arithmetic sequence. First term a 5 2 and common difference d 5 3 c) 1, 2, 4, 8, 16, is not an arithmetic sequence. d) 40, 36, 32, 28, 24, is an arithmetic sequence. First term a 5 40 and common difference d 5 24 The terms go up by 1, then 2, then 3 and so on. The terms go up by 1, then 2, then 4 and so on. Call-out boxes add clarity: new for this edition You can find the rule to give the nth term of an arithmetic sequence using the first term, a, and the common difference, d. Look at the arithmetic sequence with first term a 5 3 and common difference d 5 4. Term number n Term (n 2 1) 3 4 The first term is 3. To find each term, add 4 to the previous term. The 3rd term is lots of 4. The 20th term is lots of 4. The nth term is 3 1 (n 2 1) lots of 4. Now look at the general arithmetic sequence with first term a and common difference d. Term number n Term a a 1 d a 1 2d a 1 19d a 1 (n 2 1)d The first term is a. To find each term, add d to the previous term. The 20th term is a 1 19 lots of d. The nth term is a 1 (n 2 1) lots of d. The nth term of an arithmetic sequence is a 1 (n 2 1)d where a is the first term and d is the common difference. This material covers the new Edexcel IGCSE Higher objective 3.1 B know and use nth term = a + (n 1)d EXAMPLE a) Find a formula for the nth term of the arithmetic sequence: 7, 10, 13, 16, 19, b) Find the 50th term of the sequence. c) The nth term of the sequence is 205. Find the value of n. 14

15 9.3 Arithmetic sequences SOLUTION a) a 5 7 and d 5 3 nth term is a 1 (n 2 1)d nth term (n 21) (n 2 1) n n 1 4 b) Substitute n 5 50 into the formula for the nth term. So the 50th term c) 3n n n 5 67 So the 67th term is 205. EXERCISE The first five terms in an arithmetic sequence are: 12, 17, 22, 27, 32, a) Find the value of the 10th term. b) Write down, in terms of n, an expression for the nth term of this sequence. 2 The first four terms in an arithmetic sequence are: 58, 50, 42, 34, a) Find the value of the first negative term. b) Write down, in terms of n, an expression for the nth term of this sequence. Here are some arithmetic sequences. For each one, find, in terms of n, an expression for the nth term of the sequence. 3 8, 11, 14, 17, 20, 4 2, 7, 12, 17, 22, 5 10, 9, 8, 7, 6, 6 4, 9, 14, 19, 24, 7 21, 24, 27, 30, 33, 8 12, 10, 8, 6, 4, 9 Nina has been making patterns with sticks. Here are her first three patterns. Pattern 1 4 sticks Pattern 2 7 sticks Pattern 3 10 sticks a) Work out the number of sticks in pattern 6. b) Write down, in terms of n, an expression for the nth term of this sequence. c) Explain how the coefficients in your formula are related to the way the sticks fit together. 15

16 Chapter 9: Number sequences 10 The tenth term of an arithmetic sequence is 68 and the eleventh term is 75. a) Write down value of the common difference for this sequence. b) Work out the value of the first term. c) Write down, in terms of n, an expression for the nth term of this sequence. Check that your formula works when n 5 10 and n Arithmetic series Write the series out forwards then backwards. When you add terms of a sequence it is called a series. So 1, 4, 7, is an arithmetic sequence is an arithmetic series. There is a neat method you can use to add the terms in an arithmetic series. Look at this method of adding all the numbers from 1 to Add together the sequences You now have 100 terms which are all the same! The sum of both series is So the sum of the numbers from 1 to 100 is ½ You can use the same method to find the sum of the terms, S n, of any arithmetic series with first term a and common difference d. Write the series out forwards S n 5 a 1 (a 1 d) 1 (a 1 2d) 1 1 (a 1 (n 2 1) d) S n 5 (a 1 (n 2 1) d) 1 (a 1 (n 2 2) d) 1 (a 1 (n 2 3) d ) 1 1 a then backwards. 2 3 S n 5 2a 1 (n 21 )d 1 2a 1 (n 21) d 1 2a 1 (n 21) d 1 1 2a 1 (n 2 1) d So you have n terms which are all the same; the sum of these terms is: 2 3 S n 5 n 3 (2a 1 (n 2 1)d This is the sum of two identical series, so you need to halve it: S n 5 n (2a 1 (n 2 1)d ) 2 You will be given this formula in the exam. This material covers the new Edexcel IGCSE Higher objective 2.2 E use algebra to support and construct proofs. 16

17 9.4 Arithmetic series EXAMPLE The first term of an arithmetic series is 40 and the common difference is 2½. Find the sum of the first 21 terms of the arithmetic series. SOLUTION Substitute a 40, d 2½ and n 21 into S n S ( (21 2 1) 3 (2 1 2 )) n 2 (2a 1 (n 2 1)d ) ( (2 1 2 )) ( ) 735 Some problems are more complicated be prepared to solve simultaneous equations when tackling questions on arithmetic sequences and series. EXAMPLE The 6th term of an arithmetic series is 20. The 11th term of the same arithmetic series is 35. Find the sum of the first 100 terms of this arithmetic series. SOLUTION The nth term of an arithmetic series is a 1 (n 2 1)d. The 6th term is 20, so a 1 5d 20 and the 11th term is 35, so a 1 10d 35 You now have two equations and two unknowns, so you can solve them simultaneously to find a and d. a 1 10d 35 a 1 5d 20 Subtract to eliminate a. 5d 15 so d 3 Now substitute d 3 into a 1 5d 20 a so a 5 You can use the formula for the sum of a series, substitute a 5, d 3 and n 100 into n S n (2a 1 (n 2 1)d ) 2 S ( ( ) 3 3) ( )

18 Chapter 9: Number sequences EXERCISE Find the sum of the first 20 terms of each of the following series. a) first term is 5, common difference is 3 b) first term is 3, common difference is 5 c) first term is 3, common difference is 25 d) first term is 5, common difference is 23 New Exercise to cover new specification objectives. 2 The first three terms in an arithmetic sequence are 7, 9, 11. Find a) an expression for the nth term of the sequence b) the 20th term c) the sum of the first 50 terms. 3 Find the sum of the whole numbers from 1 to Find the sum of the first 100 multiples of 3. 5 The first term in an arithmetic series is 3. The sum of the first three terms is 21. a) Find the common difference. The last term is 99. b) How many terms are in the series? c) Find the sum of all the terms of the series. Questions 5 8 involve problem solving. 6 The 12th term of an arithmetic series is 62. The 20th term of the series is 102. Find the sum of the first 20 terms of this arithmetic series. 7 The 6th term of an arithmetic series is 20. The 11th term of the series is 35. Find the sum of the first 100 terms of this arithmetic series. 8 The second term of an arithmetic series is 95 and the fourth term is 91. a) Find the first term and the common difference. b) Find the sum of the first 100 terms. c) S n is the sum of the first n terms. What is the maximum value of S n? REVIEW EXERCISE 9 Find the next three terms in each of these number sequences. For those that form arithmetic sequences, write down, in terms of n, an expression for the nth term of this sequence. Review exercises test understanding for the chapter as a whole. 1 11, 22, 33, 44, 55, 2 2, 4, 8, 16, 32, 3 2, 5, 8, 11, 4 1, 4, 9, 16, 25, 5 10, 9, 8, 7, 6, 6 100, 99, 97, 94, 90, 7 A number sequence is defined as follows: The first term is 7. To get each new term, multiply the previous one by 3 and subtract 15. Work out the first four terms of this sequence. 18

19 Review exercise 9 8 The nth term of a number sequence is given by the expression n2 1 3n. 2 a) Work out the first five terms of this sequence. b) Do the first five terms form an arithmetic sequence? 9 Timothy has been drawing patterns. Here are his first three patterns. Pattern 1 6 sticks Pattern 2 11 sticks Pattern 3 16 sticks a) Write down the number of sticks in pattern 5. b) Work out the number of sticks in pattern 12. c) Write down, in terms of n, an expression for the nth term of this sequence. 10 Find i) the nth term and ii) the sum of the first 20 terms of each of the following arithmetic series. a) first term is 9, common difference is 4 b) first term is 4, common difference is 9 c) first term is 90, common difference is 4 d) first term is 50, common difference is 9 11 The first three terms in an arithmetic sequence are 15, 19, 24. Find a) an expression for the nth term of the sequence b) the 20th term c) the sum of the first 50 terms. New questions for this edition. 12 Find the sum of the first 50 multiples of The first term in an arithmetic series is 3. The sum of the fifth term and the sixth term is 42. a) Find the common difference. The last term is 499. b) How many terms are in the series? c) Find the sum of all the terms of the series. 14 The 11th term of an arithmetic series is 55. The 21st term of the series is 50. Find the first term and the sum of the first 100 terms of this arithmetic series. 15 Here are the first five terms of a sequence. 30, 29, 27, 24, 20, Past-paper questions a) Write down the next two terms in the sequence. are indicated. Here are the first five terms of a different sequence. 1, 5, 9, 13, 17, b) Find, in terms of n, an expression for the nth term of the sequence. [Edexcel] 19

20 Chapter 9: Number sequences 16 Here are the first five numbers of a simple sequence. 1, 5, 9, 13, 17 a) Write down the next two numbers of the sequence. b) Write down, in terms of n, an expression for the nth term of this sequence. [Edexcel] 17 Here are the first five terms of an arithmetic sequence. 6, 11, 16, 21, 26 Find an expression, in terms of n, for the nth term of this sequence. [Edexcel] A summary of essential points from the chapter. 1 Common number sequences include the positive integers, the even numbers and the odd numbers. Others you should learn to recognise are: Square numbers 1, 4, 9, 16, 25, 36, Cube numbers 1, 8, 27, 64, 125, 216, Powers of 2 1, 2, 4, 8, 16, 32, Powers of 10 1, 10, 100, 1000, , , Triangular numbers 1, 3, 6, 10, 15, 21, 2 A term-to-term rule is a rule which explains how to find the next term in a sequence using the term before. 3 A position-to-term rule is a rule that gives an expression for the nth term of a sequence. 4 In an arithmetic sequence there is a constant difference between successive terms. 5 The nth term of an arithmetic sequence is a + (n 1)d where a is the first term and d is the common difference. 6 A series is the sum of a sequence. 5, 7, 9, is an arithmetic sequence is an arithmetic series. 7 The sum S n of the first n terms of an arithmetic series is S n = n (2a + (n 1)d) 2 where a is the first term and d is the common difference. 20

21 Internet Challenge 9 Internet Challenge 9 Fibonacci numbers Fibonacci numbers are used to model the behaviour of living systems. Fibonacci numbers also lead to the Golden Ratio, widely used in classical art and architecture. In this challenge you will need to use a spreadsheet at first, before looking on the internet to complete your work. Here is the Fibonacci number sequence: 1, 1, 2, 3, 5, 8, 13, 21,. 1 Type these numbers into a computer spreadsheet, such as Excel. (It is a good idea to enter them in a vertical list, rather than a horizontal one.) 2 Each term (apart from the first two) is found by adding together the two previous ones, for example, Use your spreadsheet replicating functions to automatically generate a list of the first 50 Fibonacci numbers. 3 Divide each Fibonacci number by the one before it, for example Set up a column on your spreadsheet to do this up to the 50th Fibonacci number. What do you notice? The quantities you found in question 3 approach a limit called the Golden Ratio, f. 4 Using your spreadsheet value for f, calculate 1 2 f and 1.What do you notice? f Now use the internet to help answer the following questions. Find pictures where appropriate. riate. 5 How was the Golden Ratio used by the builders of the Parthenon in Athens? Links to mathematics 6 Whose painting of The Last Supper was based on Golden Ratio constructions? outside of the classroom. 7 Which painter was said to have attacked every canvas by the golden section? 8 When was Fibonacci born? When did he die? 9 Is there a position-to-term rule for Fibonacci numbers, that is, is there a formula for finding the nth number? 10 What sea creature has a spiral shell that is often (mistakenly) said to be based on a Golden Ratio spiral? Each chapter concludes with an Internet Challenge, for students to work through either at school or at home. This problem solving or research activity frequently goes beyond the strict boundaries of the IGCSE specification, providing enrichment and leading to a deeper understanding of mainstream topics. The Challenges may look at the history of mathematics and mathematicians, or the role of mathematics in the real world. When doing these, it is hoped that students will not just answer the written questions, but also take the time to explore the subject a little deeper. 21

22 Chapter 9: Number sequences Digital textbooks to suit today s teaching and learning styles Student etextbook We are working towards endorsement of our Student etextbook Freedom to study independently anytime, anywhere. Student etextbooks provide a downloadable version of the printed textbook that you can assign to students so they can: Download and view on any device or browser Add, edit and synchronise notes across 2 devices Access their personal copy on the move Whiteboard etextbook Whiteboard etextbooks are online, interactive versions of the printed textbooks that enable you to: Display pages for whole-class teaching Add notes and highlight areas Insert double-page spreads into your lesson plans and homework activities Ideal for whole-class discussion and annotation, Whiteboard etextbooks boost student engagement during teacher-led sessions. Student etextbook October per student for 1 year s access per student for 2 year s access per student for 3 year s access Whiteboard etextbook September (small institution up to 900 students) 250 (large institution 901+ students) To sign up for free, no obligation 30-day etextbook trials or place an order, visit 22

23 9.3 Arithmetic sequences International GCSE (9-1) Mathematics Practice for Edexcel Third Edition Endorsed for Edexcel Provide structured support and extra practice with a wealth of problem-solving and exam-style questions, in this fully updated edition written by an author team experienced in teaching and examining. Enables students to maximise their grade potential and develop their exam skills with 700 exam-style questions. Supports you and your students through the new specifications with over 100 additional questions addressing every part of the new syllabus such as the increase in Algebra. Helps build problem-solving and mathematical reasoning skills with a dedicated chapter covering all topics and subject areas, with new questions to challenge the most able students. Offers support for Higher tier students during their course and when revising for exams. Authors: Trevor Johnson was a senior examiner for a leading awarding body. He is an experienced author and former Head of Maths in a Staffordshire comprehensive school. Tony Clough is currently a senior examiner for a leading awarding body. He was Head of Maths for 22 years and has written extensively for Key Stage 3, GCSE and A Level. Edexcel International GCSE (9-1) Mathematics Practice Book Third Edition September To sign up for free, no obligation 30-day etextbook trials or place an order, visit 23

24 Chapter 9: Number sequences Dynamic Learning INTERNATIONAL GCSE (9 1) This sample chapter is taken from Edexcel International GCSE (9-1) Mathematics Student Book Third Edition, which has been selected for the Edexcel endorsement process. Ensure complete coverage of the latest Edexcel International GCSE (9-1) Mathematics A Specification, with this fully updated and highly popular Student Book written by a team experienced in examining and teaching. Supports you and your students through the new specifications with plenty of worked examples and practice questions that follow the type of problems students are likely to face in their final exam. Builds problem-solving skills with a range of new questions to test students learning and understanding. Offers a complete package of support with free downloadable resources that include detailed worked examples and answers. Introduces students to content with key objectives and a starter activity at the beginning of each chapter. Mathematics for Edexcel Specification A Our Student Books are also available in two digital formats: The Whiteboard etextbooks are online, interactive version of the books, ideal for displaying to your class and lesson planning. The Student etextbooks are downloadable versions of the books that can be assigned to students so they can view and edit a personal copy on any of their devices. FIRST TEACHING FROM SEPTEMBER 2017 THIRD EDITION Authors: Sophie Goldie is an experienced author for GCSE and A level Mathematics. She has written extensively for Hodder Education, and has been involved in versioning for many different specifications. Alan Smith was head of Maths for 15 years and responsible for introducing Edexcel s IGCSE onto the curriculum where he worked. He has published extensively and has worked as an examiner with a number of awarding bodies. Sign up for a free, no obligation trial at Textbook subject to change based on Ofqual feedback. 24 To request Inspection Copies or einspection Copies and pre-order your class sets visit