2 EXPRESSIONS AND SEQUENCES

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Neptune was the fi rst 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 identifi ed 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

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

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.

generate terms of a sequence using term-to-term and position-to-term defi nitions

derive and use the nth term of a sequence.

Chapter 2 Expressions and sequences

16 algebraic expressions expression term collecting like terms

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 simplifi ed further. 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 1.3 for use of BIDMAS).

Key Points

2.1 Collecting like terms

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.

Simplify1. a a a a 2. 4c c 5c 3. 3p2 5p2 4p2

Waitresses use algebra to note people’s orders and then collect like terms to make the order simple for the chef.

Get Ready

Objectives Why do this?

Simplify the expression 4p 2q 1 3p 5q.

4p 2q 1 3p 5q 4p 3p 2q 5q 1 p 3q 1

Alfi e is n years old. Bilal is 3 years older than Alfi e. Carla is twice as old as Alfi e.Write down an expression, in terms of n, for the total of their ages in years.Give your answer in its simplest form.

Alfi e n years Bilal (n 3) yearsCarla 2n years

Total n (n 3) 2n n n 3 2n n n 2n 3 4n 3 years

Example 1

Example 2

2 5 3so 2q 5q 3q

4p 3p 1p which is wri� en as just p.

This can be wri� en as 3 n.

This is a correct, un-simplifi ed expression.

Remove the brackets.

This is in its simplest form.

This can be wri� en as 2 n or n2 or 2n.

Examiner’s Tip

Rewrite each expression with the like terms next to each other.

M02_EMHA_SB_GCSE_0839_U02.indd 16 18/10/11 14:10:31

17substitute evaluate

2.2 Using substitution

1 Simplifya 5x 2x 3y y b 3w 7w 4z 2z c 3p q p 4q d 4a 3b a 2b e c 2d 5c 4d f 3m 7n m 4ng 5e 3f e 4f h 2x 8y 3 2y 5 i 3p q 2 5p 4q 7 j 9 a 2b 5a 4 3b

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. 4x � 2y

10y � x

2x � 5yWrite down an expression, in terms of x and y, for the perimeter of this triangle.Give your answer in its simplest form.

Exercise 2A

A03

Questions in this chapter are targeted at the grades indicated.

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.

Key Point

2.2 Using substitution

Given the value of each letter in an expression, you can work out the value of the expression by substitution.

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.

Objective

In your science lessons you need to be able to substitute into formulae when carrying out many calculations.

Why do this?

Get Ready

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

a 4a 3b 4 5 3 (3) 20 9 11b a 2b 8 5 2 (3) 8 5 6 8 3c 2a2 4b 2 (5)2 4 (3) 2 25 12 50 12 38

Example 3

Positive negative negative.

Work out the multiplication fi rst (BIDMAS). Negative negative positive.

It is only the value of a (5) that is squared.

Examiner’s Tip

Replace each letter with its numerical value.

Chapter 2 Expressions and sequences

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1 Work out the value of each of these expressions when x 4 and y 1.a x 3y b x y c 2x 5y 3 d 4x 1 2y

2 Work out the value of each of these expressions when p 2, q 3 and r 5.a p q r b 2q 3r 5p c 2q r 3pd 6 q 2r p e 5p 3q2 f p2 2q2 r2

Exercise 2B

Example 4

You can use the laws of indices to simplify algebraic expressions. See Section 1.5 for the index laws.

Key Point

a Simplify c3 c4

b Simplify 5y3z5 2y2z

a c3 c4 c c c c c c c c7

b 5y 3z5 2y 2z 5 y 3 z 5 2 y 2 z

5 2 y 3 y 2 z 5 z1

10 y 32 z 51

10 y 5 z 6

10y 5z 6

Using x p x q x p q

2.3 Using the index laws

Get Ready

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.

1. Write as a power of a single number.

a 43 48 b 78 74

______ 75 c (63)2

Watch Out!

Group like terms together before attempting to use the laws of indices.

Watch Out!

z is the same as z1

Note: 3 4 7.

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1 Simplifya m m m m m b 2p 3p c q 4q 5q

2 Simplifya a4 a7 b n n3 c x5 x d y2 y3 y4

3 Simplifya 2p2 6p4 b 4a 3a4 c b7 5b2 d 3n2 6n

4 Simplifya 5t3u2 4t5u3 b 2xy3 3x5y4 c a2b5 7a3b

d 4cd5 2cd4 e 2mn2 3m3n2 4m2n

Exercise 2C

a Simplify d5 d2

b Simplify 10x2y5

______ 2xy3

a d5 d2 d5 __

d2 d d d d d _______________ d d

d3

b 10x2y5

_______ 2xy3 is the same as 10x2y5 2xy3

10x2y5 2xy3 (10 2) (x2 x) (y 5 y 3)

5 x21 y 53

5 x y 2

5xy2

Example 5

Using x p x q x p q

1 Simplifya a7 a4 b b5 b c c

8 __

c5 d d4 d3

2 Simplifya 6q5 3q3 b 12p7 4p2 c 8x6 2x5 d

20y8

____ 2y

3 Simplifya 15a5b6 3a3b2 b 30p3q4 6p2q c 8c4d7

_____ 2c2d3 d 6x3 2x4 ________ 4x2

e 5m2n 4mn2 ___________ 2mn2

Exercise 2D

Note: 5 2 3Examiner’s Tip

Write fractions, such as p5

__ p3

as p5 p3.

2.3 Using the index laws

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Chapter 2 Expressions and sequences

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Simplify (2c3d)4

Method 1(2c3d)4 (2)4 (c3)4 (d)4

16 c3 4 d1 4

16 c12 d4

16c12d4

Method 2(2c3d)4 can be wri� en as 2c 3d 2c 3d 2c 3d 2c3dd

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

16c12d 4

Example 6

Using x p x q x p q

Using (x p)q x p q

1 Simplifya (a7)2 b (b3)5 c (c 3)3 d (d 2)8

2 Simplifya (2p3)2 b (3q2)4 c (5x 4)2 d (� m4

___ 2 ) 33 Simplify

a (2x 3y2)4 b (7e5f 3)2 c (5p 5q)3 d (� 2x4y2

_____ 3xy4 ) 3

Exercise 2E

2.4 Fractional and negative powers

Simplify these expressions.1. (a3)6 2. (3y5)3 3. (� 4a3b2

_____ 2a2b5 ) 2

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

Objective Why do this?

Examiner’s Tip

You must apply the power to number terms as well as the algebraic terms.

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The laws of indices used so far can be used to develop two further laws.x4 x4 x44 x0

Alsox4 x4 1 since any term divided by itself is equal to 1.Therefore x0 1

In generalx0 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 xp xq xp q

x 1 _ 2 x

1 _ 2 x 1 _ 2

1 _ 2 x1 xand so, x

1 _ 2 √__

x

Also, x 1 _ 3 x

1 _ 3 x 1 _ 3 x, showing that x

1 _ 3 3 √__

x

In general x

1 _ n n √__

x

Key Points

x3 x4

x x x ____________ x x x x 1 __

x

Also, using xp xq xpq

x3 x4 x34 x1

Therefore x1 1 __ x

In generalxm 1 ___

xm

Simplify (3x4y)2

(3x4y)2 1 _______ (3x 4y)2

1 _____ 9x8y2

Example 7

Using xm 1 ___ xm

Using (xp)q xp q

1 Simplifya a1 b (b2)1 c c2 d (d 3)1

2 Simplifya (e 3)2 b (f 2)4 c (x1)2 d (y1)1

3 Simplifya (x2y7)0 b (2x4y5)0 c (5p2q4)1 d (3c3d)3

e (� 2p3q ____ 3r2 ) 2

Exercise 2F

Examiner’s Tip

Remember that a negative power just means ‘one over’ or ‘the reciprocal of’.

2.4 Fractional and negative powers

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Chapter 2 Expressions and sequences

22 sequence rule terms of the sequence

Simplify (8x 6y 4 ) 1 _ 3

(8x 6y 4 ) 1 __ 3 8

1 __ 3 (x 6 ) 1 __ 3 (y 4 )

1 __ 3

3 √__

8 x 6 1 __ 3 y 4 1 __ 3

2 x2 y 4

_ 3

2x 2 y 4

_ 3

Example 8

Using x 1 __ n n √

__ x

Using (x p)q x p q

1 Simplifya (9a4 )

1 _ 2 b (16c2 ) 1 _ 4 c (27e3f 9 )

1 _ 3 d (100x3y5 ) 1 _ 2

2 Simplifya (a4 ) 1 _ 2 b (8c3 ) 1 _ 3 c (32x9y5 ) 1 _ 5 d (x2y6 ) 1 _ 4

Exercise 2G

A sequence is a pattern of shapes or numbers which are connected by a rule (or defi nition of the sequence). The relationship between consecutive terms describes the rule which enables you to fi nd 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

Key Points

2.5 Term-to-term and position-to-term de� nitions

Continue these number patterns.1. 2, 4, 6, 8, 10, … 2. 4, 9, 14, 19, 24, 29, … 3. 1, 3, 5, 7, 9, …

You can generate terms of a sequence using term-to-term and position-to-term defi nitions of the sequence.

Objective

To recognise world trends in specifi c illnesses, patterns linking data are often used.

Why do this?

Get Ready

Examiner’s Tip

Remember that the denominator of the index is the root.

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23term-to-term rule position-to-term rule

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, … The odd numbers form a sequence, 1, 3, 5, … The even numbers form a sequence, 2, 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 defi nition of the

sequence: the position-to-term rule.

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 5th term1 4 9 16

3 5 7

a The diff erence between the 4th and the 5th term is 9 and so the 5th term is 16 9 25.

b The 6th term 62 36, the 7th term 72 49, etc.

The 12th term 122 144.

Example 9

Find a the term-to-term rule,b the next two terms, and

c the 10th term for each of the following number sequences.

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

Exercise 2H

The diff erence between consecutive terms increases by 2.This is the term-to-term rule which enables you to fi nd subsequent terms of the sequence.

The numbers 1 (12), 4 ( 22), 9 ( 32), 16 ( 42) and 25 ( 52) are the fi rst fi ve square numbers.

In this way a term of the sequence can be found by the position of the term in the sequence.

2.5 Term-to-term and position-to-term defi nitions

Chapter 2 Expressions and sequences

24 arithmetic sequence difference zero term

An arithmetic sequence is a sequence of numbers where the rule is simply to add a fi xed number.For example, 2, 5, 8, 11, 14, … is an arithmetic sequence with the rule ‘add 3’.In this example the fi xed number is 3.

This is sometimes called the difference between consecutive terms. You can fi nd 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 fi nd out whether or not a given number is part of a sequence, and

explain why.

Key Points

2.6 The nth term of an arithmetic sequence

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, … 2. 4, 1, 2, 5, 8, … 3. 124, 118, 112, 106, 100, …

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.

Objectives

To be able to predict how many people might catch the fl u, epidemiologists need to develop a general rule.

Why do this?

Get Ready

Here are the fi rst fi ve 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 fi nd 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 fi rst term.Work out the zero term by using the diff erence of 3.Zero term 2 3 1

The nth term n diff erence zero term nth term n 3 1 3n 1

b For the 20th term, n 20When n 20, 3n – 1 3 20 1 60 1 59 So the 20th term is 59.

Example 10

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.

Inverse of 3.

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1 Write down i the difference between consecutive terms ii the zero term for each of the following arithmetic sequences.

a 0, 2, 4, 6, 8, … b 7, 3, 1, 5, 9, … c 14, 9, 4, 1, 6, …

2 Here are the fi rst fi ve 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 fi rst 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 fi rst fi ve 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 fi rst 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 fi rst fi ve terms of an arithmetic sequence:7 11 15 19 23Pat says that 453 is a term in this sequence. Pat is wrong.Explain why. Nov 2005

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Exercise 2I

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 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:

x0 1, for all values of x, xm 1 ___ xm and x

1 _ n n √__

x where m and n are integers.

A sequence is a pattern of shapes or numbers which are connected by a rule (or defi nition of the sequence). The relationship between consecutive terms describes the rule which enables you to fi nd 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 defi nition of the sequence:

the position-to-term rule. An arithmetic sequence is a sequence of numbers where the rule is simply to add a fi xed number. This is

called the difference between consecutive terms.

Chapter review

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Chapter review

Chapter 2 Expressions and sequences

26

You can fi nd 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 fi nd out whether or not a given number is part of a sequence, and

explain why.

1 Simplify a 3x 4y 2x y b m 7n 5m 3n

2 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. Defi ne 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.

3 Work out the value of each of these expressions when x 2, y 3 and z 7a 3x y b x 2y c x 3y 2z d 5xy e x2 y2 z2

4 Simplifya y y y b x 3x c z3 z5 d p p6 e 2a2 8a5

5 Simplifya a6 a3 b b9 b4 c 21p4 3p d 24x5

____ 3x2 e 16a6b3 2a5b3

6 Find a the rule b the next two terms c the 12th term for this number sequence. 102 99 96 93 90

7 Write down a the difference between consecutive terms, b the zero term of this arithmetic sequence. 3 2 7 12 17

8 Here are the fi rst 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.

9 Here are the fi rst four terms of an arithmetic sequence.5 8 11 14Is 140 a term in the sequence? You must give a reason for your answer.

Review exercise

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Exam Question Report

93% of students answered this sort of question well because they had learnt the rules for expressions involving indices.

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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

N h(h 1) _______ 2 when h number of cans high.

c If he has 200 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 _ 2 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.

Helvellyn HorseshoeGlenridding to Helvellyn via the edges (circular walk)Length: 8.5 kmTotal ascent: 800 mTime: 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.

12 Simplifya (a5)4 b (3b4)2 c (3e5f)3

13 The nth even number is 2n. Show algebraically that the sum of three consecutive even numbers is always a multiple of 6.

Nov 2008, adapted

14 The expression 6x2 y

_____ 4y3 can never take a negative value. Explain why.

15 a Simplify (� 9p4 ___ 4y2 )

1 _ 2

b Simplify (�2q3 ) 2

c Simplify (� 12xy3

_____ 3x5y ) 1 _

2

16 A 4 by 4 by 4 cube is placed into a tin of yellow paint.

When it has dried, the 64 individual cubes are examined.How many are covered in yellow paint on 0 sides, 1 side, 2 sides, 3 sides? Extension: Repeat the question for an n by n by n cube, and show that your expressions add up to n3.

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