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2 EXPRESSIONS AND SEQUENCES O T LL I T D S E I S E PL R - UP I H S BE Neptune was the first planet to be found by mathematical prediction. Scientists looked at the number patterns of the orbits of the other planets in the Solar System and correctly predicted Neptune’s position to within a degree. Using the predicted position, Johann Galle identified Neptune almost immediately on 23 September 1846! Objectives In this chapter you will: distinguish the different roles played by letter symbols in algebra and use the correct notation in deriving algebraic expressions manipulate algebraic expressions by collecting like terms use substitution to work out the value of an expression given the value of each letter in the expression use the index laws applied to simple algebraic expressions use the index laws applied to algebraic expressions with fractional or negative powers generate terms of a sequence using term-toterm and position-to-term definitions of the sequence derive and use the nth term of a sequence. Before you start You should be able to: simplify an expression where each term is in the same letter or letters use directed numbers in calculations use index laws with numbers. 20 2.1 Letter symbols and notation 2.1 Letter symbols and notation Objective Why do this? You can distinguish the different roles played by letter symbols in algebra and use the correct notation. A company might use algebraic symbols and numbers as a shorthand method of describing the quantities of each item when doing a stock check. Get Ready Work out the value of 1. 5 11 2. 3 7 3. 1 3 9 4. 5 (2 4) 5. 4 6 6. 4 2 Simplfy 1. a a a a 2. b 2b 3b 3. 4c c 5c 4. 2xy 3xy xy 5. 3p2 5p2 4p2 Key Points 2x, 3y and 2x 3y are called algebraic expressions Each part of an expression is called a term of the expression. 2x and 3y are terms of the expression 2x 3y. When adding or subtracting expressions, different letter symbols cannot be combined. For example 2x 3y cannot be simplified further. Example 1 Mary buys five cups of tea at t pence per cup and three cups of coffee at c pence per cup. Write down an expression, in terms of t and c, for the total cost in pence. This can be written as t 5 or 5t. The cost of five cups of tea 5 t 5t pence. The cost of three cups of coffee 3 c 3c pence. Total cost 5t 3c. Watch Out! Do not try to combine the 5t and the 3c. Exercise 2A 1 Questions in this chapter are targeted at the grades indicated. Daniel has p marbles in a bag. He gives 8 marbles to Finlay. Write down an expression, in terms of p, for the number of marbles that Daniel has left in his bag. algebraic expressions term expression letter symbol 21 Chapter 2 Expressions and sequences A03 A03 D AO2 2 Candles are sold in boxes. A small box holds 4 candles. Jo buys x small boxes of candles. a Write down, in terms of x, the total number of candles in these small boxes. A large box holds 10 candles. Jo buys 3 less of the large boxes of candles than the small boxes. b Write down, in terms of x, the number of large boxes she buys. c Find, in terms of x, the total number of candles in the large boxes that Jo buys. 3 Melissa made y cakes. Stuart made 3 more cakes than Melissa. Stuart put 5 sweets on the top of each cake. Write down an expression, in terms of y, for the number of sweets that Stuart used. 4 Angela packs pencils and pens into boxes. Each box can hold either 12 pencils or 10 pens. Angela packs some boxes with pencils and some more boxes with pens. Write down an expression, for the total number of pencils and pens Angela packs. 5 This is part of a menu at Pete’s Chippy. Pete takes an order of: two meat pies, three sausages, one fish Chips and four portions of chips. Meat pie a Write down an algebraic expression that Pete could use to write down this Sausage order of food. Define each letter that you use. Fish b Mrs Smith orders x meat pies and y portions of chips. Write down an expression, in terms of x and y, for the total cost of Mrs Smith’s order. Menu £1.20 £1.80 £0.90 £2.00 2.2 Collecting like terms Objective Why do this? You can manipulate algebraic expressions by collecting like terms. Get Ready Write down algebraic expressions for the following orders at Pete’s Chippy. (Look at the menu in Exercise 4A, question 5.) 1. Two fish, one chips. 2. Three sausages, two chips. 3. Four meat pies, one sausage, one fish and four chips. Key Points The sign of a term in an expression is always written before the term. For example, in the expression 4 2x 3y the ‘’ sign means add 2x and the ‘’ sign means subtract 3y. The term x can be written as 1x. In algebra, BIDMAS describes the order of operations when collecting like terms (see Section 1.5 for use of BIDMAS). 22 collecting like terms 2.2 Collecting like terms Example 2 Simplify these expressions. a 3e 2f e 5f b g 3h 6g 7h c 4p 2q 1 3p 5q a 3e 2f e 5f 3e e 2f 5f 4e 7f Examiner’s Tip 4e and 7f are not like terms. b g 3h 6g 7h g 6g 3h 7h 7g 4h 3 7 4 so 3h 7h 4h c 4p 2q 1 3p 5q 4p 3p 2q 5q 1 p 3q 1 Rewrite each expression with the like terms next to each other. 2 5 3 so 2q 5q 3q 4p 3p 1p which is written as just p. Example 3 Alfie is n years old. Bilal is 3 years older than Alfie. Carla is twice as old as Alfie. Write down an expression, in terms of n, for the total of their ages in years. Give your answer in its simplest form. Alfie n years Bilal (n 3) years Carla 2n years This can be written as 3 n. This can be written as 2 n or n2 or 2n. Total n (n 3) 2n n n 3 2n n n 2n 3 4n 3 years This is a correct un-simplified expression. Remove the brackets. This is in its simplest form. Exercise 2B 1 Simplify a 5x 2x 3y y c 3p q p 4q e c 2d 5c 4d g 5e 3f e 4f i 3p q 2 5p 4q 7 b d f h j 3w 7w 4z 2z 4a 3b a 2b 3m 7n m 4n 2x 8y 3 2y 5 9 a 2b 5a 4 3b like terms 23 Chapter 2 Expressions and sequences A03 2 Georgina, Samantha and Mason collect football stickers. Georgina has x stickers in her collection. Samantha has 9 stickers less than Georgina. Mason has 3 times as many stickers as Georgina. Write down an expression, in terms of x, for the total number of these stickers. Give your answer in its simplest form. 3 The diagram shows a triangle. Write down an expression, in terms of x and y, for the perimeter of this triangle. Give your answer in its simplest form. 4x � 2y 2x � 5y 10y � x 2.3 Using substitution Objective Why do this? Given the value of each letter in an expression, you can work out the value of the expression by substitution. In your science lessons you need to be able to substitute into formulae when carrying out many calculations. Get Ready Write expressions, in terms of x and y, for the perimeter of these rectangles. 1. Length 2x 4, width y 2 2. Length 3y 3, width x 5 3. Length 4x 5, width y 2. Key Point If you are given the value for each letter in an expression then you can evaluate the expression. Example 4 Work out the value of each of these expressions when a 5 and b 3. a 4a 3b b a 2b 8 c 2a2 4b Examiner’s Tip a 4a 3b 4 5 3 (3) 20 9 11 Positive negative negative. b a 2b 8 5 2 (3) 8 5 6 8 3 c 2a2 4b 2 (5)2 4 (3) 2 25 12 50 12 38 24 evaluate Replace each letter with its numerical value. Work out the multiplication first (BIDMAS). Negative negative positive. It is only the value of a (5) that is squared. 2.4 Using the index laws Exercise 7C 1 2 Work out the value of each of these expressions when x 4 and y 1. a x 3y b xy c 2x 5y 3 d 4x 1 2y Work out the value of each of these expressions when p 2, q 3 and r 5. a pqr b 2q 3r 5p c 2q r 3p 2 d 6 q 2r p e 5p 3q f p2 2q2 r2 2.4 Using the index laws Objective Why do this? You understand and can use the index laws applied to simple algebraic expressions. To write large numbers, like the speed of sound, indices are often used to shorten the way the value is written. Get Ready Simplify these expressions using the index laws. 1. x3 x2 2. x6 x4 3. (x2)3 Key Points You can use the laws of indices to simplify algebraic expressions. See Section 1.6 for the index laws. Example 5 a Simplify c3 c4 Watch Out! b Simplify 5y3z5 2y2z a c3 c4 c c c c c c c c7 Group like terms together before attempting to use the laws of indices. Note: 3 4 7. b 5y 3z5 2y 2z 5 y 3 z 5 2 y 2 z z is the same as z1 5 2 y 3 y 2 z 5 z1 10 y 32 z 51 10 y 5 z 6 5 6 10y z Using x p x q x p q 25 Chapter 2 Expressions and sequences Exercise 2D D C 1 2 3 4 Simplify a mmmmm b 2p 3p c q 4q 5q Simplify a a4 a7 b n n3 c x5 x d y2 y3 y4 Simplify a 2p2 6p4 b 4a 3a4 c b7 5b2 d 3n2 6n Simplify a 5t3u2 4t5u3 b 2xy3 3x5y4 d 4cd5 2cd4 e 2mn2 3m3n2 4m2n Example 6 c a2b5 7a3b a Simplify d5 d2 10x2y5 b Simplify ______ 2xy3 5 d _______________ ddddd a d5 d2 __ d2 dd d3 Examiner’s Tip Note: 5 2 3 p5 Write fractions, such as __3 p as p5 p3. 2 5 10x y b _______ is the same as 10x2y5 2xy3 3 2xy 10x2y5 2xy3 (10 2) (x2 x) (y 5 y 3) 5 x21 y 53 5 x y2 5xy 2 Using x p x q x p q Exercise 2E C 1 2 B 3 Simplify a a7 a4 b b5 b c8 c __ c5 d d4 d3 Simplify a 6q5 3q3 b 12p7 4p2 c 8x6 2x5 20y8 d ____ 2y Simplify a 15a5b6 3a3b2 b 30p3q4 6p2q 8c4d7 c _____ 2c2d3 6x3 2x4 d ________ 4x2 5m2n 4mn2 e ___________ 2mn2 26 2.5 Fractional and negative powers Example 7 Simplify (2c3d)4 Method 1 (2c3d)4 (2)4 (c3)4 (d)4 Examiner’s Tip 16 c3 4 d1 4 Using (x p)q x p q 12 16 c 4 d 16c12d4 You must apply the power to number terms as well as the algebraic terms. Method 2 (2c3d)4 can be written as 2c 3d 2c 3d 2c 3d 2c3d 2 2 2 2 c 3 c 3 c 3 c3 d d d d 16 c 3 3 3 3 d 4 16 c12 d 4 Using x p x q x p q 16c12d 4 Exercise 2F 1 2 3 C Simplify a (a7)2 b (b3)5 c (c 3)3 d (d 2)8 Simplify a (2p3)2 b (3q2)4 c (5x 4)2 d m 2 ) (___ Simplify a (2x 3y2)4 b (7e5f 3)2 c (5p 5q)3 d (3xy ) 4 3 2x y _____ 4 2 3 B 4 2.5 Fractional and negative powers Objective Why do this? You can use the index laws applied to algebraic expressions with fractional or negative powers. To write very small numbers, like the radius of a molecule, negative powers of 10 are used. Get Ready Simplify these expressions. 1. (a3)6 2. (3y5)3 ( 4a3b2 3. _____ 2a2b5 ) 2 27 Chapter 2 Expressions and sequences Key Points The laws of indices used so far can be used to develop two further laws. x4 x4 x44 x0 x3 x4 x x x __ 1 Also ____________ xxxx x 4 4 x x 1 since any term divided by itself Also, using xp xq xpq is equal to 1 x3 x4 x34 x1 Therefore x0 1 1 Therefore x1 __ x In general In general x0 1 1 xm ___ xm The laws of indices can be used further to solve problems with fractional indices. __ The square root of x is written √x, and you know that: __ __ √x √x x Using xp xq xp q _1 _1 _1 _1 x2 x2 x2 2 x1 x _1 __ and so, x2 √x _1 _1 _1 _1 __ 3 Also, x3 x3 x3 x, showing that x3 √ x In general __ _1 xn n√x Example 8 Simplify (3x4y)2 1 (3x4y)2 _______ (3x 4y)2 1 _____ 9x8y2 1 Using xm ___ m Examiner’s Tip x Remember that a negative power just means ‘one over’ or ‘the reciprocal of’. Using (xp)q xp q Exercise 2G B 1 2 A 3 Simplify a a1 b (b2)1 c c2 d (d 3)1 Simplify a (e 3)2 b (f 2)4 c (x1)2 d (y1)1 Simplify a (x2y7)0 b (2x4y5)0 c (5p2q4)1 d (3c3d)3 e 28 (2p 3r q ) 3 ____ 2 2 2.6 Term-to-term and position-to-term definitions _1 Example 9 Simplify (8x 6y 4)3 1 __ 1 __ 1 __ 1 __ (8x 6y 4)3 83 (x 6)3 (y 4)3 3 __ √8 x 6 __13 n __ 1 __ Using x n √ x Examiner’s Tip 4 __13 y Remember that the denominator of the index is the root. Using (x p)q x p q _4 2 x2 y 3 _4 2x 2y 3 Exercise 2H 1 2 Simplify _1 a (9a4)2 b (16c ) Simplify _1 a (a4) 2 b (8c3) 3 2 _1 d (100x y ) _1 d (x2y6) 4 _1 c (27e f _1 c (32x9y5) 5 4 ) 3 9 3 5 3 _1 A 2 _1 2.6 Term-to-term and position-to-term definitions Objective Why do this? You can generate terms of a sequence using term-to-term and position-to-term definitions of the sequence. To recognise trends in specific illnesses in a country or the world, patterns linking data are often used. Get Ready Continue these number patterns. 1. 2, 4, 6, 8, 10, … 2. 4, 9, 14, 19, 24, 29, … 3. 1, 3, 5, 7, 9, … Key Points A sequence is a pattern of shapes or numbers which are connected by a rule (or definition of the sequence). The relationship between consecutive terms describes the rule which enables you to find subsequent terms of the sequence. Here is a sequence of 4 square patterns made up of squares: Pattern 1 Pattern 2 Pattern 3 Pattern 4 sequence rule terms of the sequence 29 Chapter 2 Expressions and sequences Each pattern above is a term of the sequence; is the 1st term in the sequence, is the 2nd term in the sequence, etc. The number of squares in each term form a sequence of numbers, 1, 4, 9, 16, … You can continue a sequence if you know how the terms are related: the term-to-term rule. You can continue a sequence if you know how the position of a term is related to the definition of the sequence: the position-to-term rule. Example 10 Find a the next term, and b the 12th term of the sequence of numbers: 1, 4, 9, 16, … 1st term 2nd term 3rd term 4th term 1 4 9 16 3 5 5th term 7 a The difference between the 4th and the 5th term is 9 and so the 5th term is 16 9 25. The difference between consecutive terms increases by 2. This is the term-to-term rule which enables you to find subsequent terms of the sequence. b The 6th term 62 36, the 7th term 72 49, etc. The numbers 1 (12), 4 ( 22), 9 ( 32), 16 ( 42) and 25 ( 52) are the first five square numbers. The 12th term 122 144. In this way a term of the sequence can be found by the position of the term in the sequence. Exercise 2I Find a the term-to-term rule, b the next two terms, and c the 10th term for each of the following number sequences. 30 1 2 5 8 11 2 4 2 8 14 3 19 12 5 2 4 1 3 6 10 5 0 2 6 12 term-to-term position-to-term 2.7 The nth term of an arithmetic sequence 2.7 The nth term of an arithmetic sequence Objectives Why do this? You can use linear expressions to describe the nth term of a sequence. You can use the nth term of a sequence to generate terms of the sequence. To be able to predict how many people might catch flu, epidemiologists need to develop a general rule. Get Ready Find 1. the rule, a 1, 4, 7, 10, … 2. the next two terms, b 4, 1, 2, 5, 8, … 3. the 10th term for each of the following number sequences. c 124, 118, 112, 106, 100, … Key Points An arithmetic sequence is a sequence of numbers where the rule is simply to add a fixed number. For example, 2, 5, 8, 11, 14, … is an arithmetic sequence with the rule ‘add 3’. In this example the fixed number is 3. This is sometimes called the difference between consecutive terms. You can find the nth term using the result nth term n difference zero term. You can use the nth term to generate the terms of a sequence. You can use the terms of a sequence to find out whether or not a given number is part of a sequence, and explain why. Example 11 Here are the first five terms of an arithmetic sequence: 2, 5, 8, 11, 14, … a Write down, in terms of n, an expression for the nth term of the arithmetic sequence. b Use your answer to part a to find the 20th term. zero term 1st term 2nd term 3rd term 4th term 5th term –1 2 5 8 11 14 3 3 3 3 3 difference a The zero term is the term before the first term. Work out the zero term by using the difference of 3. Zero term 2 3 1 Inverse of 3. The nth term n difference zero term nth term n 3 1 3n 1 b For the 20th term, n 20 When n 20, 3n – 1 3 20 1 60 1 59 So the 20th term is 59. Examiner’s Tip Always check your answer by substituting values of n into your nth term. For example, 1st term, when n 1, 3n 1 3 1 1 2 ✓ 2nd term, when n 2, 3n 1 3 2 1 5 ✓ 3rd term, when n 3, 3n 1 3 3 1 8 ✓ etc. arithmetic sequence difference zero term 31 Chapter 2 Expressions and sequences Exercise 2J C 1 Write down a the difference between consecutive terms b the zero term for each of the following arithmetic sequences. 0, 2, 4, 6, 8, … 7, 3, 1, 5, 9, … 14, 9, 4, 1, 6, … AO3 AO3 2 Here are the first five terms of an arithmetic sequence: 1, 7, 13, 20, 26, … a Write down, in terms of n, an expression for the nth term of this arithmetic sequence. b Use your answer to part a to work out the i 12th term, ii 50th term. 3 Here are the first four terms of an arithmetic sequence: 7, 11, 15, 19, … a Write down, in terms of n, an expression for the nth term of this arithmetic sequence. b Use your answer to part a to work out the i 15th term, ii 100th term. 4 Here are the first five terms of an arithmetic sequence: 32, 27, 22, 17, 12, … a Write down, in terms of n, an expression for the nth term of this arithmetic sequence. b Use your answer to part a to work out the i 20th term, ii 200th term. 5 Here are the first four terms of an arithmetic sequence: 18, 25, 32, 39, … Explain why the number 103 cannot be a term of this sequence. 6 Here are the first five terms of an arithmetic sequence: 7 11 15 19 23 Write down, in terms of n, an expression for the nth term of this sequence. Pat says that 453 is a term in this sequence. Pat is wrong. Explain why. Nov 2005 Chapter review 2x, 3y and 2x 3y are called algebraic expressions. Each part of an expression is called a term of the expression. When adding or subtracting expressions, different letters cannot be combined. The sign of a term in an expression is always written before the term. The term x can be written as 1x. In algebra, BIDMAS describes the order of operations. If you are given the value for each letter in an expression then you can evaluate the expression. You can use the laws of indices to simplify algebraic expressions. The basic index laws can be used to develop further laws: xp xq xp q xp xq xp q (xp)q xp q and x0 1, for all values of x 1 and x_n1 n√__ and xm ___ x where m and n are integers. xm A sequence is a pattern of shapes or numbers which are connected by a rule (or definition of the sequence). 32 Chapter review The relationship between consecutive terms describes the rule which enables you to find subsequent terms of the sequence. You can continue a sequence if you know how the terms are related: the term-to-term rule. You can continue a sequence if you know how the position of a term is related to the definition of the sequence: the position-to-term rule. An arithmetic sequence is a sequence of numbers where the rule is simply to add a fixed number. This is called the difference between consecutive terms. The nth term of an arithmetic sequence can be found using the result nth term n difference zero term. You can find the nth term of an arithmetic sequence using the result nth term n difference zero term. You can use the nth term of an arithmetic sequence to generate the terms of a sequence. You can use the terms of a sequence to find out whether or not a given number is part of a sequence, and explain why. Review exercise 1 A cup of tea costs x pence. A cup of coffee costs y pence. Viv buys 3 cups of tea and 5 cups of coffee. Write down an expression, in terms of x and y, for the total cost. 2 Samantha is absent from school on n days. Georgina is absent from school on 3 more days than Samantha. Melissa is absent from school on 5 days less than Georgina. Write down, in terms of n, the number of days that Melissa is absent from school. 3 Simplify 4 Helen and Stuart collect stamps. Helen has 240 British stamps and 114 Australian stamps. a Write down an algebraic expression that could be used to represent Helen’s British and Australian stamps. Define the letters used. Stuart has 135 British stamps and 98 Australian stamps. b Using the same letters, write down an algebraic expression that could be used to represent the total of Helen’s and Stuart’s British and Australian stamps. 5 To calculate the cost of printing leaflets for a school fair, the printer uses the formula: C 40 0.05n where C is the cost in pounds and n is the number of leaflets printed. a How much would it cost to print 200 leaflets? a 3x 4y 2x y b m 7n 5m 3n b Can you suggest what 40 and 0.05 represent? 6 7 Work out the value of each of these expressions when x 2, y 3 and z 7 a 3x y b x 2y c x 3y 2z d 5xy D 2 2 2 e x y z The formula used to convert temperatures in Fahrenheit, F, into Celsius, C, is given by: 5(F – 32) C _______ 9 a Find C when F 77. b Use the formula to find the freezing point of water in Fahrenheit. A newspaper headline read ‘Phew, what a scorcher! Temperature soars into the 100s.’ c What temperature unit are they using? What is its equivalent in the other unit? AO2 AO3 33 Chapter 2 Expressions and sequences C 8 9 AO3 AO3 AO3 AO3 AO2 AO3 Simplify a yyy b x 3x c z3 z5 Simplify a a6 a3 b b9 b4 c 21p4 3p b the next two terms 90 d p p6 e 2a2 8a5 24x5 d ____ e 16a6b3 2a5b3 3x2 c the 12th term for this number sequence. 10 Find a the rule 102 99 96 93 11 Write down 12 Here are the first five terms of an arithmetic sequence: 0, 4, 8, 12, 16, ... a Write down, in terms of n, an expression for the nth term of this arithmetic sequence. b Use your answer to part a to work out the i 20th term, ii 1000th term. 13 Here are the first four terms of an arithmetic sequence: 204, 192, 180, 168, … a Write down, in terms of n, an expression for the nth term of this arithmetic sequence. b Use your answer to part a to work out the i 13th term ii 99th term. 14 Here are the first five terms of an arithmetic sequence: 3, 7, 11, 15, 19, ... Prove that the number 5381 is not a term in this sequence. 15 The nth term of a sequence is n2 4 Alex says “The nth term of the sequence is always a prime number when n is an odd number.” Is Alex correct? You must give a reason for your answer. Nov 2008, Paper 2 Adapted 16 Here are the first 5 terms of a sequence. 1 1 2 3 5 The rule for the sequence is “The first two terms are 1 and 1. To get the next term add the two previous terms” Explain why, after the first two terms, the other terms of the sequence are alternately even and odd. 17 Here are the first four terms of an arithmetic sequence. 5 8 11 14 Is 140 a term in the sequence? You must give a reason for your answer. 18 Neal is asked to produce an advertising stand for a new variety of soup. He stacks the cans according to the pattern shown. a the difference between consecutive terms, b the zero term of this arithmetic sequence. 3 2 7 12 17 The stack is 4 cans high and consists of 10 cans. a How many cans will there be in a stack 10 cans high? h(h 1) b Verify that the total number of cans (N) can be calculated by the formula N _______ when 2 n number of cans high. c If he has 200 cans, how high can he make his stack? 34 Chapter review 19 Naismith, an early Scottish mountain climber, devised a formula that is still used today to calculate how long it will take mountaineers to climb a mountain. The metric version states: Allow one hour for every 5 km you walk forward and add on _12 hour for every 300 m of ascent. a How long should it take to walk 20 km with 900 m of ascent? A mountain walker’s guide contains the following information for a particular walk. AO3 C Helvellyn Horseshoe Glenridding to Helvellyn via the edges (circular walk) Length: 8.5 km Total ascent: 800 m Time: 4 hour round trip b Calculate how long this walk should take according to Naismith’s formula. Give your answer to the nearest minute. c Suggest reasons why this time is different to the one in the guidebook. 20 21 B Simplify a (a5)4 b (3b ) c (3e f) 4 2 5 3 The n th even number is 2n Show algebraically that the sum of three consecutive numbers is always a multiple of 6. AO3 Nov 2008, Paper 4 Adapted 22 23 24 6x y The expression _____ can never take a negative value. Explain why 4y3 2 AO3 A Simplify a (x5) 1 b (y ) Simplify _1 a (16g6)2 b (64xy6)3 c (xy) (4y ) 2 _1 _1 1 4 _2 d (2m n) 0 3 6 _1 c (y8)2 25 26 A 4 by 4 by 4 cube is placed into a tin of yellow paint. b Simplify (2q ) 2 d (a4b10)4 (3x y ) 1 3 _2 12xy c Simplify _____ 5 9p a Simplify ___2 3 4 AO3 When it has dried, the 64 individual cubes are examined. How many are covered in yellow paint on 0 sides, 1 side, 2 sides ...? Extension 1: Repeat the question for an n by n by n cube, and show that your expressions add up to n3. Extension 2: Repeat the question for a cuboid of dimensions l by m by n. ?? Jim has £x. Bill has £3 more than than Jim. Beccy has twice as much as Bill. Write down an expression, in terms of x, for the total amount in pounds they have altogether. 35
