2 EXPRESSIONS AND SEQUENCES

Transcription

1 EXPRESSIONS AND SEQUENCES Neptune was the first planet to be found by mathematical prediction. Scientists looked at the number patterns of the orbits of 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 September 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 collect like terms use substitution to work out the value of an expression use the index laws applied to simple algebraic expressions and to algebraic expressions with fractional or negative powers generate terms of a sequence using term-toterm and position-to-term definitions derive and use the nth term of a sequence. Before you start You need to be able to: simplify an expression where each term is in the same unknown or unknowns use directed numbers in calculations use index laws with numbers. 15

2 Chapter Expressions and sequences.1 Collecting like terms Objectives You can distinguish the different roles played by letter symbols in algebra and use the correct notation. You can manipulate algebraic expressions by collecting like terms. Why do this? Waitresses use algebra to note people s orders and then collect like terms to make the order simple for the chef. Get Ready Simplify 1. a a a a. 4c c 5c. p 5p 4p Key Points x, y and x y are called algebraic expressions. Each part of an expression is called a term of the expression. x and y are terms of the expression x y. When adding or subtracting expressions, different letter symbols cannot be combined. For example x y cannot be simplified further. The sign of a term in an expression is always written before the term. For example, in the expression 4 x y the sign means add x and the sign means subtract y. The term x can be written as 1x. In algebra, BIDMAS describes the order of operations when collecting like terms (see Section 1. for use of BIDMAS). Example 1 Simplify the expression 4p q 1 p 5q. 4p q 1 p 5q 4p p q 5q 1 p q 1 5 so q 5q q 4p p 1p which is written as just p. Rewrite each expression with the like terms next to each other. Example Alfie is n years old. Bilal is 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 ) years Carla n years Total n (n ) n n n n n n n 4n years This can be written as n. This can be written as n or n or n. This is a correct, un-simplified expression. Remove the brackets. This is in its simplest form. 16 algebraic expressions expression term collecting like terms

3 . Using substitution Exercise A Questions in this chapter are targeted at the grades indicated. 1 Simplify a 5x x y y b w 7w 4z z c p q p 4q d 4a b a b e c d 5c 4d f m 7n m 4n g 5e f e 4f h x 8y y 5 i p q 5p 4q 7 j 9 a b 5a 4 b Georgina, Samantha and Mason collect football stickers. Georgina has x stickers in her collection. Samantha has 9 stickers less than Georgina. Mason has 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. 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 y 10y x x 5y A0. Using substitution Objective Given the value of each letter in an expression, you can work out the value of the expression by substitution. Why do this? 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 x 4, width y. length y, width x 5. length 4x 5, width y. Key Point If you are given the value for each letter in an expression then you can substitute the values into the expression and evaluate the expression. Example Work out the value of each of these expressions when a 5 and b. a 4a b b a b 8 c a 4b a 4a b 4 5 ( ) b a b 8 5 ( ) c a 4b (5) 4 ( ) Positive negative negative. Work out the multiplication first (BIDMAS). Negative negative positive. Replace each letter with its numerical value. It is only the value of a ( 5) that is squared. substitute evaluate 17

4 Chapter Expressions and sequences Exercise B 1 Work out the value of each of these expressions when x 4 and y 1. a x y b x y c x 5y d 4x 1 y Work out the value of each of these expressions when p, q and r 5. a p q r b q r 5p c q r p d 6 q r p e 5p q f p q r. Using the index laws Objective You understand and can use the index laws applied to simple algebraic expressions. Why do this? To write large numbers, like the speed of sound, indices are often used to shorten the way the value is written. Get Ready 1. Write as a power of a single number. a b c (6 ) Key Point You can use the laws of indices to simplify algebraic expressions. See Section 1.5 for the index laws. Example 4 a Simplify c c 4 b Simplify 5y z 5 y z a c c 4 c c c c c c c c 7 Note: 4 7. Watch Out! Group like terms together before attempting to use the laws of indices. b 5y z 5 y z 5 y z 5 y z z is the same as z 1 5 y y z 5 z 1 10 y z y 5 z 6 Using x p x q x p q 10y 5 z 6 18

5 . Using the index laws Exercise C 1 Simplify a m m m m m b p p c q 4q 5q Simplify a a 4 a 7 b n n c x 5 x d y y y 4 D C Simplify a p 6p 4 b 4a a 4 c b 7 5b d n 6n 4 Simplify a 5t u 4t 5 u b xy x 5 y 4 c a b 5 7a b d 4cd 5 cd 4 e mn m n 4m n Example 5 a Simplify d 5 d b Simplify 10x y 5 xy a d 5 d d5 d d d d d d d d b d 10x y 5 xy is the same as 10x y 5 xy Note: 5 Write fractions, such as p5 p as p 5 p. 10x y 5 xy (10 ) (x x) (y 5 y ) 5 x 1 y 5 5 x y Using x p x q x p q 5xy Exercise D 1 Simplify a a 7 a 4 b b 5 b c c 8 c 5 d d4 d Simplify a 6q 5 q b 1p 7 4p c 8x 6 x 5 d Simplify a 15a 5 b 6 a b b 0p q 4 6p q c e 5m n 4mn mn 0y 8 y 8c 4 d 7 c d d 6x x 4 4x C B 19

6 Chapter Expressions and sequences Example 6 Simplify (c d) 4 Method 1 (c d) 4 () 4 (c ) 4 (d) 4 16 c 4 d c 1 d 4 16c 1 d 4 Using (x p ) q x p q You must apply the power to number terms as well as the algebraic terms. Method (c d) 4 can be written as c d c d c d c dd c c c c d d d d 16 c d 4 16 c 1 d 4 Using x p x q x p q 16c 1 d 4 Exercise E C B 1 Simplify a (a 7 ) b (b ) 5 c (c ) d (d ) 8 Simplify a (p ) b (q ) 4 c (5x 4 ) d ( m4 Simplify ) a (x y ) 4 b (7e 5 f ) c (5p 5 q) d ( x4 y xy 4 ).4 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. (a ) 6. (y 5 ). ( 4a b a b 5 ) 0

7 A.4 Fractional and negative powers Key Points The laws of indices used so far can be used to develop two further laws. x 4 x 4 x 4 4 x 0 x x 4 Also x x x x 4 x 4 x x x x 1 x 1 since any term divided by itself Also, using x p x q x p q is equal to 1. x x 4 x 4 x 1 Therefore x 0 1 In general x 0 1 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 x p x q x p q x 1_ x 1_ x 1_ 1_ x 1 x and so, x 1_ x Also, x 1_ x 1_ x 1_ x, showing that x 1_ x In general x 1_ n n x Therefore x 1 1 x In general x m 1 x m Example 7 Simplify (x 4 y) (x 4 y) 1 (x 4 y) 1 9x 8 y Using x m 1 x m Using (x p ) q x p q Remember that a negative power just means one over or the reciprocal of. Exercise F 1 Simplify a a 1 b (b ) 1 c c d (d ) 1 B Simplify a (e ) b (f ) 4 c (x 1 ) d (y 1 ) 1 Simplify a (x y 7 ) 0 b (x 4 y 5 ) 0 c (5p q 4 ) 1 d (c d) e ( p q r ) 1

8 Chapter Expressions and sequences Example 8 Simplify (8x 6 y 4 ) 1_ (8x 6 y 4 ) (x 6 ) 1 (y 4 ) 1 8 x 6 1 y 4 1 x y 4_ x y 4_ Using x 1 n n x Using (x p ) q x p q Remember that the denominator of the index is the root. Exercise G A 1 Simplify a (9a 4 ) 1_ b (16c ) 1_ 4 c (7e f 9 ) 1_ d (100x y 5 ) 1_ Simplify a (a 4 ) 1_ b (8c ) 1_ c (x 9 y 5 ) 1_ 5 d (x y 6 ) 1_ 4.5 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 world trends in specific illnesses, patterns linking data are often used. Get Ready Continue these number patterns. 1., 4, 6, 8, 10,. 4, 9, 14, 19, 4, 9,. 1,, 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 Pattern Pattern 4 sequence rule terms of the sequence

9 .5 Term-to-term and position-to-term definitions Each pattern above is a term of the sequence; is the 1st term in the sequence, is the nd term in the sequence, etc. The number of squares in each term form a sequence of numbers, 1, 4, 9, 16, The odd numbers form a sequence, 1,, 5, The even numbers form a sequence,, 4, 6, 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 9 Find a the next term, and b the 1th term of the sequence of numbers: 1, 4, 9, 16, 1st term nd term rd term 4th term 5th term a The difference between the 4th and the 5th term is 9 and so the 5th term is The difference between consecutive terms increases by. This is the term-to-term rule which enables you to find subsequent terms of the sequence. b The 6th term 6 6, the 7th term 7 49, etc. The numbers 1 ( 1 ), 4 ( ), 9 ( ), 16 ( 4 ) and 5 ( 5 ) are the first five square numbers. The 1th term In this way a term of the sequence can be found by the position of the term in the sequence. Exercise H Find a the term-to-term rule, b the next two terms, and c the 10th term for each of the following number sequences term-to-term rule position-to-term rule

10 Chapter Expressions and sequences.6 The nth term of an arithmetic sequence Objectives 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. Why do this? To be able to predict how many people might catch the flu, epidemiologists need to develop a general rule. Get Ready Find a the rule, b the next two terms, c the 10th term for each of the following number sequences. 1. 1, 4, 7, 10,. 4, 1,, 5, 8,. 14, 118, 11, 106, 100, Key Points An arithmetic sequence is a sequence of numbers where the rule is simply to add a fixed number. For example,, 5, 8, 11, 14, is an arithmetic sequence with the rule add. In this example the fixed number is. 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 10 Here are the first five terms of an arithmetic sequence:, 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 0th term. zero term 1st term nd term rd term 4th term 5th term difference a The zero term is the term before the first term. Work out the zero term by using the difference of. Zero term 1 Inverse of. The nth term n difference zero term nth term n 1 n 1 b For the 0th term, n 0 When n 0, n So the 0th term is 59. Always check your answer by substituting values of n into your nth term. For example, 1st term, when n 1, n nd term, when n, n rd term, when n, n etc. 4 arithmetic sequence difference zero term

11 Chapter review Exercise I 1 Write down i the difference between consecutive terms ii the zero term for each of the following arithmetic sequences. a 0,, 4, 6, 8, b 7,, 1, 5, 9, c 14, 9, 4, 1, 6, C * Here are the first five terms of an arithmetic sequence: 1, 7, 1, 0, 6, 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 1th term, ii 50th term. 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:, 7,, 17, 1, 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 0th term, ii 00th term. 5 Here are the first four terms of an arithmetic sequence: 18, 5,, 9, Explain why the number 10 cannot be a term of this sequence. 6 Here are the first five terms of an arithmetic sequence: Pat says that 45 is a term in this sequence. Pat is wrong. Explain why. Nov 005 AO AO Chapter review x, y and x y are called algebraic expressions. Each part of an expression is called a term of the expression. When adding or subtracting expressions, different letter symbols 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 when collecting like terms. If you are given the value for each letter in an expression then you can substitute the values into the expression and 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: x 0 1, for all values of x, x m 1 1_ n m and x n x where m and n are integers. x 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. 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. 5

12 Chapter Expressions and sequences 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 Simplify a x 4y x y b m 7n 5m n Helen and Stuart collect stamps. Helen has 40 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 15 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. D C Work out the value of each of these expressions when x, y and z 7 a x y b x y c x y z d 5xy e x y z 4 Simplify a y y y b x x c z z 5 d p p 6 e a 8a 5 5 Simplify a a 6 a b b 9 b 4 c 1p 4 p d 4x 5 x e 16a6 b a 5 b Exam Question Report 9% of students answered this sort of question well because they had learnt the rules for expressions involving indices. 6 Find a the rule b the next two terms c the 1th term for this number sequence Write down a the difference between consecutive terms, b the zero term of this arithmetic sequence Here are the first four terms of an arithmetic sequence: 04, 19, 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 1th term ii 99th term. AO 9 Here are the first four terms of an arithmetic sequence Is 140 a term in the sequence? You must give a reason for your answer. 6

13 A Chapter review 10 Neal is asked to produce an advertising stand for a new variety of soup. He stacks the cans according to the pattern shown. The stack is 4 cans high and consists of 10 cans. a How many cans will there be in a stack 10 cans high? b Verify that the total number of cans (N) can be calculated by the formula h(h 1) N when h number of cans high. c If he has 00 cans, how high can he make his stack? 11 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 1_ hour for every 00 m of ascent. a How long should it take to walk 0 km with 900 m of ascent? A mountain walker s guide contains the following information for a particular walk. AO AO AO 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. 1 Simplify a (a 5 ) 4 b (b 4 ) c (e 5 f) B 1 The nth even number is n. Show algebraically that the sum of three consecutive even numbers is always a multiple of 6. Nov 008, adapted 14 The expression 6x y can never take a negative value. Explain why. 4y 1_ 15 a Simplify ( 9p4 4y ) 1 x 5 y ) b Simplify ( q ) c Simplify ( 1xy 16 A 4 by 4 by 4 cube is placed into a tin of yellow paint. AO AO AO When it has dried, the 64 individual cubes are examined. How many are covered in yellow paint on 0 sides, 1 side, sides, sides? Extension: Repeat the question for an n by n by n cube, and show that your expressions add up to n. 7