13 SEQUENCES 1assets.pearsonschool.com/asset_mgr/versions/2012-05/AE19...13 SEQUENCES 1.1 Section title In this chapter you will: continue number patterns using the four rules of number

1.1 Section title13 SEQUENCES

In this chapter you will: continue number patterns using the four rules of

number continue patterns using pictures and give the

rule for continuing the pattern use number machines to produce a number

pattern and write down the rule complete a table of values using a number

machine and write down the rule use the fi rst difference to fi nd the nth term

and use the nth term to fi nd any number in a sequence

identify whether or not a number is in a sequence.

You need to know: how to spot how a number pattern continues that the numbers in number patterns

can get bigger and get smaller

about odd numbers about even numbers that multiples are members of a multiplication

table, e.g. 3, 6, 9, 12 are multiples of 3.

Neptune was the fi rst planet to be found by mathematical prediction. It was found by looking at the number patterns of the other planets in the Solar System, and its position was correctly predicted to within a degree. Two scientists were eventually jointly credited with the discovery, one British and one French, but it has since been shown that the Brits took a bit too much of the credit!

Objectives Before you start

LOW RES PIC

241

Chapter 13 Sequences

242 terms term to term rules sequence

Key Points

A sequence is a pattern of numbers or shapes that follows a rule. Number patterns can be continued by adding, subtracting, multiplying and dividing. Patterns with pictures can be continued by fi nding the rule for continuing the pattern. The numbers in a number pattern are called terms. The term to term rules for continuing number patterns can be given. (This means you can say how you fi nd a

term from the one before it.)

13.1 Sequences

You can continue number patterns by adding or subtracting, or multiplying or dividing by a number.

You can give the term to term rules for continuing number patterns.

You can continue patterns using pictures and give the rule for continuing the pattern.

Get Ready

Objectives

You may use sequences when learning a dance routine or a note sequence when playing a musical instrument.

Why do this?

Even numbers form a pattern 2, 4, 6, 8, 10, 12, ... They go up in twos.Odd numbers also form a pattern 1, 3, 5, 7, 9, 11, ... These also go up in twos.

1. Write down all the even numbers up to 20.2. Write down all the odd numbers up to 20.3. Check that you have written all the numbers from 1 to 20.

Continuing patterns by adding

a Write down the next two numbers in this number pattern.2 6 10 14

b What is the rule you use to fi nd the next number in the number pattern?c Find the 10th number in this pattern.

a 14 4 18 18 4 22

b To get the next number you add 4 each time.

c The 10th number in the pa� ern is 38.

Example 1

2 6 10

�4 �4

14 18

�4 �4

2 6 10 14 18 22 26 30 34 38Carry on the number pa� ern until you get to the 10th number in the pa� ern.

243

13.1 Sequences

Key Point

Number patterns can be continued by subtracting the same number from each term.

Continuing patterns by subtracting

a Write down the next two numbers in this number pattern.60 54 48 42 36


a 36 6 30 30 6 24

Example 2

60 54 48

�6 �6

42 36

�6 �6

To get to the next number you take away 6.Take away 6 from 36 to get 30 then take away 6 from 30 to get 24.

1 Find the two missing numbers in these number patterns. For each pattern, write down the term to term rule.a 3, 6, 9, __ , __ , 18, 21 b 3, 7, 11, __ , __ , 23, 27c 5, 10, 15, 20, __ , __ , 35, 40 d 2, 7, 12, 17, __ , __ , 32, 37e 1, 4, 7, 10, __ , __ , 19, 22 f 5, 7, 9, 11, __ , __ , 17, 19g 3, 8, 13, 18, __ , __ , 33, 38 h 4, 7, 10, 13, __ , __ , 22, 25i 2, 6, 10, 14, __ , __ , 26, 30 j 10, 20, 30, __ , __ , 60, 70

2 a Write down the next two numbers in these sequences. i 1, 5, 9, 13, 17, … ii 2, 5, 8, 11, 14, … iii 3, 7, 11, 15, 19, … iv 4, 8, 12, 16, 20, … v 5, 8, 11, 14, 17, … vi 5, 11, 17, 23, … vii 2, 6, 10, 14, 18, … viii 1, 7, 13, 19, 25, … ix 3, 11, 19, 27, 35, … x 5, 9, 13, 17, 21, …

b Write down the rule you used to fi nd the missing numbers in each sequence.

3 Find the 10th number of each of the number patterns in questions 1 and 2.

4 Jenny saves £2 each week in her piggy bank. Here is the pattern of how her money grows.

Week 1 2 3 4 5Money in piggy bank 2 4 6

a Copy and complete the table.b Jenny is saving for a present for her Mum’s birthday that costs £20.

How many weeks will this take?

Exercise 13A Questions in this chapter are targeted at the grades indicated.

Examiner’s Tip

… means that the sequence carries on.

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b To get the next number you subtract 6 each time.


60 54 48 42 36 30 24 18Carry on the number pa� ern until you get to the 8th number in the pa� ern.

1 Find the two missing numbers in these number patterns. Write down the rule for each number pattern.a 20, 18, 16, 14, __ , __ , 8 b 17, 15, 13, 11, __ , __ , 5c 55, 50, 45, 40, __ , __ , 25 d 42, 37, 32, 27, __ , __ , 12e 22, 19, 16, 13, __ , __ , 4 f 19, 17, 15, 13, __ , __ , 7g 45, 38, 31, 24, __ , __ , 3 h 25, 22, 19, 16, __ , __ , 7i 29, 25, 21, 17, __ , __ , 5 j 80, 70, 60, __ , __ , 30

2 a Write down the next two numbers in these sequences. i 41, 37, 33, 29, … ii 27, 24, 21, 18, … iii 59, 55, 51, 47, … iv 34, 31, 28, 25, … v 30, 27, 24, 21, … vi 61, 55, 49, 43, … vii 22, 20, 18, 16, … viii 51, 46, 41, 36, … ix 64, 57, 50, 43, … x 8, 6, 4, 2, 0, 2, …

b Write down the rule you used to fi nd the missing numbers in each sequence.


4 Abdul’s mother gives him £20 each week to buy his lunch. His lunch costs him £3 each day. Here is the pattern of how he spends his money.

Day M Tu W Th FMoney left at end of day 17 14

a Copy and complete the table.

b How much money will Abdul have left at the end of the week?

Exercise 13B

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13.1 Sequences

Continuing number patterns by multiplying

a Write down the next two numbers in this number pattern.1 3 9 27 81


a 81 3 243 243 3 729

b To get the next number you multiply by 3 each time.


Example 3

1 3 9

�3 �3

27 81

�3 �3To get to the next number you multiply by 3.Multiply 81 by 3 to get 243 then multiply 243 by 3 to get 729.

1 3 9 27 81 243 729 2187Carry on the number pa� ern until you get to the 8th number in the pa� ern.

1 Find the missing numbers in these number patterns. For each pattern, write down the rule.a 1, 2, 4, 8, __ , __ , 64 b 1, 4, 16, 64, __ , 1024c 1, 5, __ , 125, __ , 3125 d 1, 10, 100, __ , __ , 100 000e 3, 6, 12, 24, __ , __ , 192 f 2, 6, 18, __ , __ , 486g 2, 8, 32, __ , __ , 2048 h 2, 20, 200, 2000, __ , __ , 2 000 000i 2, 10, 50, __ , __ , 6250 j 3, 15, 75, __ , 1875

2 a Write down the next two numbers in these sequences. i 2, 4, 8, 16, … ii 3, 9, 27, 81, … iii 4, 16, 64, 256, … iv 5, 25, 125, 625, … v 5, 10, 20, 40, … vi 4, 12, 36, 108, … vii 10, 30, 90, 270, … viii 5, 50, 500, 5000, … ix 10, 20, 40, 80, … x 6, 36, 216, 1296, …

b Write down the rule you used to fi nd the missing number in each sequence.


4 The number of rabbits in a particular colony doubled every month for 10 months. The table shows the beginning of the pattern.

Month 1 2 3 4 5Number of rabbits 2 4 8

a Copy and complete the table.b How many rabbits were in the colony in month 10?

Exercise 13C

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Continuing number patterns by dividing

a Write down the next two numbers in this number pattern.729 243 81 27


a 27 3 9 9 3 3

b To get the next number you divide by 3 each time.

c The 8th number in the pa� ern is 1 3 1 __ 3

Example 4

81 27 9

�3 �3

3 1

�3 �3

To get to the next number you divide by 3.Divide 27 by 3 to get 9 then divide 9 by 3 to get 3.

729 243 81 27 9 3 1 1 __ 3 Carry on the number pa� ern until you get to the 8th number in the pa� ern.1 3 1 __ 3

1 Find the missing numbers in these number patterns. Write down the rule for each number pattern.a 64, 32, 16, 8, __ , __ , 1 b 1024, 256, 64, __ , 4c 3125, 625, 125, __ , __ , 1 d 100 000, 10 000, 1000, __ , __ , 1e 192, 96, 48, 24, __ , __ , 3 f 486, 162, 54, 18, __ , 2g 1024, 512, 256, 128, __ , __ , 16 h 300 000, 30 000, 3000, __ , __ , 3i 6250, 1250, 250, __ , __ , 2 j 2000, 200, 20, __ , __ , 0.02

2 a Write down the next two numbers in these sequences. i 64, 32, 16, 8, … ii 243, 81, 27, 9, … iii 128, 64, 32, 16, … iv 625, 125, 25, 5, … v 80, 40, 20, 10, … vi 972, 324, 108, 36, … vii 2430, 810, 270, 90, … viii 50 000, 5000, 500, 50, … ix 160, 80, 40, 20, … x 1296, 216, 36, 6, …

b Write down the rule you used to fi nd the missing number in each sequence.


4 The number of radioactive atoms in a radioactive isotope halves every 10 years. The table shows the beginning of the pattern.

Years 0 10 20 30 40Number of atoms 2560 1280 640

a Copy and complete the table.b How many radioactive atoms were in the isotope in year 100?

Exercise 13D

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13.1 Sequences

Continuing patterns in pictures

a Copy and complete the table for the number of matches used to make each member of the pattern.

Pattern number 1 2 3 4 5 6 7Number of matches used 4 7 10

b Write down the rule to get the next number in the pattern.c How many matches are there in pattern number 10?

a

10 3 13 13 3 1616 3 19 19 3 22

b Add 3 to the previous number.

c Pa� ern number 10 has 31 matches.

Example 5

Pa� ern number

1 2 3 4 5 6 7

Number of matches used

4 7 10 13 16 19 22 4 7 10

�3 �3 �3 �3

Continue the pa� erns.22 25 28 31

Count the number of matches in each pa� ern and write down the number of matches used.

1 For these patterns: i draw the next two patterns ii write down the rule in words to fi nd the next pattern iii use your rule to fi nd the 10th term.

a

b

c

d

e

Exercise 13E

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13.2 Using input and output machines to investigate number patterns

You can use number machines to produce a number pattern and write down the rule (term number to term).

You can complete a table of values using a number machine and write down the rule (term number to term).

You can fi nd missing values in a table of values and use the term number to term rule.

Get Ready

Objectives

When baking, you take your ingredients, mix them together and bake them in the oven, and you end up with a cake.

This process applies to anything from baking a cake to making a motor car.

Why do this?

1. Put the following numbers into this number machine and write down the answers.a 5 b 3 c 11 d 17

2. Draw a number machine for the process 6 and use it to fi nd the answer when the following numbers are put into it.a 10 b 6 c 2

Input Action Output

�3

2 a Write down the number of matches in each of these patterns.

Pattern 1 Pattern 2 Pattern 3

b Draw the next two patterns.c Write down the rule in words to continue the pattern.d Use your rule to fi nd the number of matches needed for pattern number 10.

3 Repeat question 2 with the hexagon shape shown below.

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249table of values one-stage input and output machines


Key Points

In this number pattern 3 7 11 15 19 23 …Term 1 Term 2 Term 3 Term 4

3, 7, 11, 15, 19, 23, … are the terms.

The term number tells you the position of each term in the pattern.In the sequence 3, 7, 11, 15, 19, … term 1 is 3, term 2 is 7, etc.

You can use number machines to produce a number pattern and write down the rule (term number to term rule).

You can complete a table of values using a number machine and write down the rule (term number to term).

You can fi nd missing values in a table of values and use the term number to term rule.

Sometimes you can put two number machines together to make a sequence.

One-stage input and output machines

This number machine has been used to produce the terms of a pattern.

a Complete the term numbers and terms in this table of values for the number machine.

Term number Term

1 5234

b What is the rule for working out the term from the term number?c Write down the rule for fi nding the next term from the term before it.

a Term number Term Term number Term1 5 1 5

52 10 2 10

53 15 3 15

54 20 4 20

b Multiply the term number by 5.

c Add 5.

Example 6�5

The rule for the number machine is multiply the term number by 5 so the terms will be 5, 10, 15, 20.

To get to the term from the term number you multiply by 5.

To get to the next term from the term before it you have to add 5 since the pa� ern is 5, 10, 15, 20, …


250

For each of these questions:a copy and complete the table of values for the number machineb write down the rule for fi nding the term from the term numberc write down the rule for fi nding the next term from the term before it.

1

�3

2

�7

3

�4

4

�2

5

�8

6

�10

7

�12

8

�50

Exercise 13F

Term number Term1 32 634








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Two-stage input and output machines

This number machine has been used to produce the terms of a pattern.

a Complete the terms in a table of values for the number machine.

Term number Term

1 4234

b What is the rule for working out the term from the term number?

c Write down the rule for fi nding the next term from the term before it.

a Term number

TermTerm

numberTerm Working

1 4 1 4 1 3 1 43

2 7 2 7 2 3 1 73

3 10 3 10 3 3 1 103

4 13 4 13 4 3 1 13

b Multiply by 3 and add 1.

c Add 3.

Example 7�3Term number Term�1

To get to the term from the term number you 3 and 1.

To get to the next term from the term before it you have to add 3 since the pa� ern is 4, 7, 10, 13, …

Examiner’s Tip

Use the rule from the number machine on the term number to get to the term. You feed the result of the fi rst machine into the second machine.


252

For each of these questions:a copy and complete the table of values for the number machineb write down the rule for fi nding the term from the term numberc write down the rule for fi nding the next term from the term before it.

1 �3 �2

2 �2 �1

3 �3 �1

4 �4 �3

5 �5 �2

6 �3 �4

7 �3 �5

8 �5 �1

9 �4 �3

10 �5 �4

Term number 1 2 3 4 5

Term


Term


Term


Term


Term


Term


Term


Term


Term


Term

Exercise 13G

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Complete this table of values for the number pattern with term number to term rule, ‘Multiply by 4 and subtract 2’.

Example 8

Term number

Term

1 2

2 6

3

4

5

↓ ↓

8

↓ ↓

38

4 2

Term number

Term Working

1 2 1 4 2 22 6 2 4 2 63 10 3 4 2 104 14 4 4 2 145 18 5 4 2 18↓ ↓

8 30 8 4 2 30↓ ↓

10 38 10 4 2 38

You can fi nd these terms by using the rule 4 then 2.

Examiner’s Tip

Don’t forget Bidmas: you do the before the .You met Bidmas in Chapter 8.

Copy and complete these tables of values.

1 2 3

Exercise 13H

Term number

Term

1 42345↓ ↓

10↓ ↓

34

3 1

Term number

Term

1 12345↓ ↓

10↓ ↓

25

2 1

Term number

Term

1 82345↓ ↓

10↓ ↓

78

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254 fi rst difference

13.3 Finding the nth term of a number pattern

You can use the fi rst difference to fi nd the nth term of a number pattern and use the nth term to fi nd any number in a number pattern or sequence.

Get Ready

Objective

This may be useful when your teacher is dividing the class into groups, so that you can work out which group you are going to be in, or make sure you will be in a group with your friends.

Why do this?

1. Write down the difference between each term in these number patterns.a 5, 10, 15, 20, 25, 30, … b 40, 35, 30, 25, 20, … c 4, 7, 10, 13, 16, …d 7, 11, 15, 19, 21, … e 50, 47, 44, 41, 37, …

2. Find the 10th term in each of the number patterns in question 1.

Key Point

The fi rst difference can be used to fi nd the nth term of a number pattern and then the nth term can be used to fi nd any number in a sequence.

4 5 6

7 a Find the 10th number in this number pattern. 3, 7, 11, 15, …b What is the term number for the term that is 47?



Term number

Term

1 12345↓ ↓

10↓ ↓

45

4 3

Term number

Term

1 112345↓ ↓

10↓ ↓

151

10 1

Term number

Term

1 22345↓ ↓

10↓ ↓

67

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13.3 Finding the nth term of a number pattern

Here is a number pattern 4, 7, 10, 13, 16, …a Find the nth term in this pattern.b Find the 20th term in this number pattern.

a Term number

Term Diff erence

1 43

3

3

3

2 73 104 135 16

n 3n 1

b The 20th term is 61.

Example 9

Step 4Compare your new pa� ern with the original one and see what number you need to add or subtract to/from each term to get the original number pa� ern. In this case it is 1.The nth term is 3n 1.You replace the n by 20 in the nth term to fi nd the 20th term. It is 3 20 1 61

Step 1Put the number pa� ern into a table of values.

Step 2Find the diff erence between the terms in the number pa� ern. In this case it is 3.

Step 3Multiply each term number by the diff erence to get a new pa� ern.3, 6, 9, 12, 15 …

1 For questions 1, 2 and 3 in Exercise 10H, fi nd the nth term of each of the number patterns.

2 Write each pattern in a table and use the table to fi nd the nth term of these number patterns.Use your nth term to fi nd the 20th term in each of these number patterns.

a 1, 3, 5, 7, 9, 11, … b 3, 5, 7, 9, 11, 13, …

c 2, 5, 8, 11, 14, 17, … d 5, 8, 11, 14, 17, 20, …

e 1, 5, 9, 13, 17, 21, … f 2, 6, 10, 14, 18, 22, …

g 2, 7, 12, 17, 22, 27, … h 4, 9, 14, 19, 24, 29, …

i 8, 13, 18, 23, 28, … j 5, 7, 9, 11, 13, …

k 40, 35, 30, 25, 20, … l 38, 36, 34, 32, 30, …

m 35, 32, 29, 26, 23, … n 20, 18, 16, 14, 12, …

o 19, 17, 15, 13, 11, … p 190, 180, 160, 150, …

Exercise 13I

Examiner’s Tip

To fi nd the nth term of a sequence that gets smaller you subtract a multiple of n from a fi xed number.e.g. 15 2n is the nth term of 13, 11, 9, 7, …

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13.4 Deciding whether or not a number is in a number pattern

You can use number patterns or use the nth term to identify whether a number is in the pattern.

Get Ready

Objective

This is useful when you want to work out what will happen in the future, for example, you could work out whether next year will be a leap year as this happens every four years.

Why do this?

1. Write each of these patterns in a table and use the table to fi nd the nth term.Use your nth term to fi nd the 20th term in each pattern.

a 4, 7, 10, 13, … b 3, 8, 13, 18, … c 13, 15, 17, 19, …

Key Points

Number patterns or the nth term can be used to identify whether a number is in the pattern.

Sometimes you will be asked how you know if a number is part of a sequence. You would then have to explain why the number is in the sequence or, even, why it is not in the sequence.

3 Here is a pattern made from sticks.

Pattern number 1 Pattern number 2 Pattern number 3

a Draw pattern number 4.

b Copy and complete this table of values for the number of sticks used to make the patterns.

Pattern number 1 2 3 4 5 6

Number of sticks 6 10

c Write, in terms of n, the number of sticks needed for pattern number n.

d How many sticks would be needed for pattern number 20?

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Chapter review

Here is a number pattern.3 8 3 18 23a Explain why 423 is in the pattern.b Explain why 325 is not in the pattern.

a 423 is in the number pa� ern.Every odd term ends in 3 and goes up 3, 13, 23, etc, so 423 will be a member as it ends in a 3.

b 325 is not in the number pa� ern.325 ends in a 5 and every member of the pa� ern ends in either a 3 or an 8 so 325 cannot be in the pa� ern.

Example 10

A sequence is a number or shape pattern which follows a rule. Number patterns can be continued by adding, subtracting, multiplying and dividing. The term to term rules for continuing number patterns can be given. Patterns using pictures can be continued by fi nding the rule for continuing the pattern. You can use number machines to produce a number pattern and write down the rule (term number to term rule) You can complete a table of values using a number machine and write down the rule (term number to term). You can fi nd missing values in a table of values and use the term number to term rule. The fi rst difference can be used to fi nd the nth term of a number pattern and then the nth term can be used

to fi nd any number in a sequence. Number patterns or the nth term can be used to identify whether a number is in the pattern.

There are other ways of answering questions like these. For example, you could identify the nth termThe nth term is 5n 2 if 5n 2 423 5n 425 so n 85so 423 is the 85th term.

The nth term is 5n 2 so if 325 is in the pa� ern 5n 2 325 5n 327 so n 65.4If 325 is in the pa� ern n must be a whole number.65.4 is not a whole number so 325 is not in the pa� ern.

For each of these number patterns, explain whether each of the numbers in brackets are members of the number pattern or not.

1 1, 3, 5, 7, 9, 11, … (21, 34) 2 3, 5, 7, 9, 11, 13, … (63, 86)

3 2, 5, 8, 11, 14, 17, … (50, 66) 4 5, 8, 11, 14, 17, 20, … (50, 62)

5 1, 5, 9, 13, 17, 21, … (101, 150) 6 2, 6, 10, 14, 18, 22, … (101, 98)

7 2, 7, 12, 17, 22, 27, … (97, 120) 8 4, 9, 14, 19, 24, 29, … (168, 169)

9 40, 35, 30, 25, 20, … (85, 4) 10 38, 36, 34, 32, 30, … (71, 82)

11 3, 7, 11, 15, 19, 21, … (46, 79) 12 5, 11, 17, 23, 29, … (119, 72)

Exercise 13J

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1 Here are some patterns made of squares.

Pattern number 1 Pattern number 2 Pattern number 3

The diagram below shows part of Pattern number 4.a Copy and complete Pattern number 4

Pattern number 4

b Find the number of squares used for Pattern number 10 November 2008 adapted

2 Here are the fi rst 4 terms in a number sequence.

124 122 120 118

a Write down the next term in this number sequence.b Write down the 7th term in this number sequence.c Can 9 be a term in this number sequence? You must give a reason for your answer. May 2009

3 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. November 2008 adapted

4 Here are the fi rst 5 terms of a sequence.

1 1 2 3 5

The rule for the sequence is ‘The fi rst two terms are 1 and 1. To get the next term add the two previous terms’.a Find the 6th term and the 7th term.b Find the 10th term.c Explain why after the fi rst two terms the other terms of the sequence are alternately even and odd.

The rule for another sequence is ‘The fi rst two terms are 2 and 2. To get the next term multiply the two previous terms’.d Find an expression for the 10th term of this sequence. You do not have to wok out the expression.

5 Here are the fi rst four terms of an arithmetic sequence.

5 8 11 14

Is 140 a term in the sequence? You must fi ve a reason for your answer

6 The fi rst term of a sequence is x. To get the next term, multiply the previous term by 2 and add 1.The third term of the sequence is 21. Find the value of x.

Review exercise

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13 SEQUENCES 1assets.pearsonschool.com/asset_mgr/versions/2012-05/AE19...13 SEQUENCES 1.1 Section title In this chapter you will: continue number patterns using the four rules of number