year 8 sequences

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A4 Sequences

KS3 Mathematics


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AA4.1 Introducing sequences

Contents

A4 Sequences

A4.2 Describing and continuing sequences

A4.3 Generating sequences


4, 8, 12, 16, 20, 24, 28, 32, ...

In maths, we call a list of numbers in order a sequence.

Each number in a sequence is called a term.

If terms are next to each other they are referred to as consecutive terms.

When we write out sequences, consecutive terms are usually separated by commas.

1st term 6th term

Introducing sequences


A sequence can be infinite. That means it continues forever.

For example, the sequence of multiples of 10,

Infinite and finite sequences

10, 20, 30, 40, 50, 60, 70, 80, 90

is infinite. We show this by adding three dots at the end.

, ...

If a sequence has a fixed number of terms it is called a finite sequence.

For example, the sequence of two-digit square numbers

16, 25 ,36, 49, 64, 81

is finite.


Some sequences follow a simple rule that is easy to describe.

For example, this sequence

Sequences and rules

2, 5, 8, 11, 14, 17, 20, 23, 26, 29, …

continues by adding 3 each time. Each number in this sequence is one less than a multiple of three.

Other sequences are completely random.

For example, the sequence of winning raffle tickets in a prize draw.

In maths we are mainly concerned with sequences of numbers that follow a rule.


Here are the names of some sequences which you may know already:

2, 4, 6, 8, 10, ...

1, 3, 5, 7, 9, ...

3, 6, 9, 12, 15, ...

5, 10, 15, 20, 25, ...

1, 4, 9, 16, 25, ...

Even Numbers (or multiples of 2)

Odd numbers

Multiples of 3

Multiples of 5

Square numbers

1, 3, 6, 10, 15, ... Triangular numbers

Naming sequences


Ascending sequences

When each term in a sequence is bigger than the one before the sequence is called an ascending sequence.

For example,

The terms in this ascending sequence increase in equal steps by adding 5 each time.

2, 7, 12, 17, 22, 27, 32, 37, ...

+5 +5 +5 +5 +5 +5 +5

The terms in this ascending sequence increase in unequal steps by starting at 0.1 and doubling each time.

0.1, 0.2, 0.4, 0.8, 1.6, 3.2, 6.4, 12.8, ...

×2 ×2 ×2 ×2 ×2 ×2 ×2


Descending sequences

When each term in a sequence is smaller than the one before the sequence is called a descending sequence.

For example,

The terms in this descending sequence decrease in equal steps by starting at 24 and subtracting 7 each time.

24, 17, 10, 3, –4, –11, –18, –25, ...

–7 –7 –7 –7 –7 –7 –7

The terms in this descending sequence decrease in unequal steps by starting at 100 and subtracting 1, 2, 3, …

100, 99, 97, 94, 90, 85, 79, 72, ...

–1 –2 –3 –4 –5 –6 –7


Number sequences are all around us.Some sequences, like the ones we have looked at today follow a simple rule.Some sequences follow more complex rules, for example, the time the sun sets each day.

Some sequences are completely random, like the sequence of numbers drawn in the lottery.

What other number sequences can be made from real-life situations?

Sequences from real-life


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Contents

A4 Sequences

A4.1 Introducing sequences



We can show many well-known sequences using geometrical patterns of counters.

Even Numbers

Odd Numbers

Sequences from geometrical patterns

2 4 6 8 10

1 3 5 7 9


Multiples of Three

Multiples of Five


3 6 9 12 15

5 10 15 20 25


Square Numbers

Triangular Numbers


1 4 9 16 25

1 3 6 10 15


How could we arrange counters to represent the sequence 2, 6, 12, 20, 30, ...?

The numbers in this sequence can be written as:

1 × 2, 2 × 3, 3 × 4, 4 × 5, 5 × 6, ...

We can show this sequence using a sequence of rectangles:

Sequences with geometrical patterns

1 × 2 = 2 2 × 3 = 6 3 × 4 = 12 4 × 5 = 20 5 × 6 = 30


Powers of two

21 = 2 22 = 4 23 = 8 24 = 16 25 = 32 26 = 64

We can show powers of two like this:

Each term in this sequence is double the term before it.


31 = 3 32 = 9 33 = 27 34 = 81 35 = 243 36 = 729

Powers of three

We can show powers of three like this:

Each term in this sequence is three times the term before it.


Sequences that increase in equal steps

We can describe sequences by finding a rule that tells us how the sequence continues.

To work out a rule it is often helpful to find the difference between consecutive terms.

For example, look at the difference between each term in this sequence:

3, 7, 11, 15, 19, 23, 27, 31, ...

+4 +4 +4 +4 +4 +4 +4

This sequence starts with 3 and increases by 4 each time.

Every term in this sequence is one less than a multiple of 4.


Can you work out the next three terms in this sequence?

How did you work these out?

This sequence starts with 22 and decreases by 6 each time.

Sequences that decrease in equal steps

22, 16, 10, 4, –2, –8, –14, –20, ...

–6 –6 –6 –6 –6 –6 –6

Each term in the sequence is two less than a multiple of 6.

Sequences that increase or decrease in equal steps are called linear or arithmetic sequences.

Sequences that increase or decrease in equal steps are called linear or arithmetic sequences.


Some sequences increase or decrease in unequal steps.

Sequences that increase in increasing steps

For example, look at the differences between terms in this sequence:

5, 6, 8, 11, 15, 20, 26, 33, ...

+1 +2 +3 +4 +5 +6 +7

This sequence starts with 5 and increases by 1, 2, 3, 4, …

The differences between the terms form a linear sequence.




This sequence starts with 7 and decreases by 0.1, 0.2, 0.3, 0.4, 0.5, …

Sequences that decrease in decreasing steps

7, 6.9, 6.7, 6.4, 6, 5.5, 4.9, 4.2, ...

–0.1 –0.2 –0.3 –0.4 –0.5 –0.6 –0.7

With sequences of this type it is often helpful to find a second row of differences.



Look at the differences between terms.

Using a second row of differences

1, 3, 8, 16, 27, 41, 58, 78, ...

+2 +5 +8 +11 +14 +17 +20

A sequence is formed by the differences so we look at the second row of differences.

+3 +3 +3 +3 +3 +3

This shows that the differences increase by 3 each time.


Some sequences increase or decrease by multiplying or dividing each term by a constant factor.

Sequences that increase by multiplying

For example, look at this sequence:

2, 4, 8, 16, 32, 64, 128, 256, ...

×2 ×2 ×2 ×2 ×2 ×2 ×2

This sequence starts with 2 and increases by multiplying the previous term by 2.

All of the terms in this sequence are powers of 2.




This sequence starts with 1024 and decreases by dividing by 4 each time.

Sequences that decrease by dividing

1024, 256, 64, 16, 4, 1, 0.25, 0.0625, ...

÷4 ÷4 ÷4 ÷4 ÷4 ÷4 ÷4

We could also continue this sequence by multiplying by each time.

14




This sequence starts with 1, 1 and each term is found by adding together the two previous terms.

Fibonacci-type sequences

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, ...

This sequence is called the Fibonacci sequence after the Italian mathematician who first wrote about it.

1 + 1 1 + 2 2 + 3 3 + 5 5 + 8 8 + 13 13 + 21 21 + 34


Describing and continuing sequences

Here are some of the types of sequence you may come across:

Sequences that increase or decrease by adding together the two previous terms.

Sequences that increase or decrease in equal steps. These are called linear or arithmetic sequences.

Sequences that increase or decrease in unequal steps by multiplying or dividing by a constant factor.

Sequences that increase or decrease in unequal steps by adding or subtracting increasing or decreasing numbers.


A number sequence starts as follows

1, 2, ...

Continuing sequences

How many ways can you think of continuing the sequence?

Give the next three terms and the rule for each one.


Finding missing terms


Name that sequence!


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Contents

A4 Sequences

A4.1 Introducing sequences



Sequence grid


Generating sequences from flow charts

A sequence can be given by a flow chart. For example,

START

Write down 3.

Add on 1.5.

Write down the answer.

Is the answer more than 10?

No

Yes

STOP

This flow chart generatesthe sequence: 3, 4.5, 6, 6.5, 9, 10.5.

This sequence has only six terms.

It is finite.


START

Write down 5.

Subtract 2.1.


Is the answer less than –3?

No

Yes

STOP


This flow chart generates the sequence 5, 2.9, 0.8, –1.3, –3.4.


START

Write down 200.

Divide by 2.


Is the answer less than 7?

No

Yes

STOP


This flow chart generates the sequence 200, 100, 50, 25, 12.5, 6.25.


START

Write down 3 and 4.

Add together the two previous numbers.


Is the answer more than 100?

No

Yes

STOP


This flow chart generates the sequence 3, 4, 7, 11, 18, 29, 47, 76, 123.


Predicting terms in a sequence

Usually, we can predict how a sequence will continue by looking for patterns.

For example, 87, 84, 81, 78, ...

We can predict that this sequence continues by subtracting 3 each time.

However, sequences do not always continue as we would expect.

For example,

A sequence starts with the numbers 1, 2, 4, ...

How could this sequence continue?


Here are some different ways in which the sequence might continue:

1

+1

2

+2

4

+3

7

+4

11

+5

16

+6

22

1

×2

2

×2

4

×2

8

×2

16

×2

32

×2

64

We can never be certain how a sequence will continue unless we are given a rule or we can justify a rule from a practical context.



This sequence continues by adding 3 each time.

We can say that rule for getting from one term to the next term is add 3.

This is called the term-to-term rule.

The term-to-term rule for this sequence is +3.


1

+3

4

+3

7

+3

10

+3

13

+3

16

+3

19


Does the rule +3 always produce the same sequence?

No, it depends on the starting number.

For example, if we start with 2 and add on 3 each time we have,

2,

17, 20, 23, ...5, 8, 11, 14,

If we start with 0.4 and add on 3 each time we have,

0.4,

15.4, 18.4, 21.4, ...3.4, 6.4, 9.4, 12.4,

Using a term-to-term rule


Writing sequences from term-to-term-rules

A term-to-term rule gives a rule for finding each term of a sequence from the previous term or terms.A term-to-term rule gives a rule for finding each term of a sequence from the previous term or terms.

To generate a sequence from a term-to-term rule we must also be given the first number in the sequence.

For example,

1st term

5

Term-to-term rule

Add consecutive even numbers starting with 2.

This gives us the sequence,

5

+2

7

+4

11

+6

17

+8

25

+10

35

+12

47 ...


Write the first five terms of each sequence given the first term and the term-to-term rule.

1st term Term-to-term rule

21

80

48

50000

–1

1.2

Sequences from a term-to-term rule

10

100

3

5

7

0.8

Add 3

Subtract 5

Double

Multiply by 10

Subtract 2

Add 0.1

10, 13, 16, 19,

100, 95, 90, 85,

3, 6, 12, 24,

5, 50, 500, 5000,

7, 5, 3, 1,

0.8, 0.9, 1.0, 1.1,


Generating sequences

Fill in the missing terms in these sequences and write down their

term to term rule.


Generating sequences

Follow the instructions in each question to produce a sequence.

year 8 sequences

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