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KS3 Mathematics

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KS3 Mathematics

A4 Sequences

A4 Sequences

Contents

A4.2 Describing and continuing sequences

A4.1 Introducing sequences

A4.3 Generating sequences

A4.4 Finding the nth term

A4.5 Sequences from practical contexts

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

1st term

6th term

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.

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

For example, the sequence of multiples of 10,

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

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, . . .

Even Numbers (or multiples of 2)

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

Odd numbers

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

Multiples of 3

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

Multiples of 5

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

Square numbers

1, 3, 6, 10,15, . . .

Triangular numbers

×2

+5

+5

+5

+5

+5

+5

+5

×2

×2

×2

×2

×2

×2

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, . . .

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, . . .

–7

–7

–7

–7

–7

–7

–7

–7

–1

–2

–3

–4

–5

–6

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, . . .

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, . . .

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

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.

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

A4 Sequences

Contents

A4.1 Introducing sequences

A4.2 Describing and continuing sequences

A4.3 Generating sequences

A4.4 Finding the nth term

A4.5 Sequences from practical contexts

2

4

6

8

10

1

3

5

7

9

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

Even Numbers

Odd Numbers

3

6

9

12

15

5

10

15

20

25

Multiples of Three

Multiples of Five

4

1

9

16

25

1

3

6

10

15

Square Numbers

Triangular Numbers

2 × 3 = 6

3 × 4 = 12

4 × 5 = 20

5 × 6 = 30

1 × 2 = 2

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:

22 = 4

23 = 8

21 = 2

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

We can show powers of three like this:

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

+4

+4

+4

+4

+4

+4

+4

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, . . .

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

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

–6

–6

–6

–6

–6

–6

–6

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

22, 16, 10, 4, –2,

–8,

–14,

–20, . . .

How did you work these out?

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

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.

+1

+2

+3

+4

+5

+6

+7

Some sequences increase or decrease in unequal steps.

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

2, 6, 8, 11, 15, 20, 26, 33, . . .

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

The differences between the terms form a linear sequence.

–0.1

–0.2

–0.3

–0.4

–0.5

–0.6

–0.7

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

7, 6.9, 6.7, 6.4, 6,

5.5,

4.9,

4.2, . . .

How did you work these out?

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

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

+20

+17

+14

+8

+5

+2

+3

+3

+3

+3

+3

+11

+3

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

1, 3, 8, 16, 27,

41,

58,

78, . . .

Look at the differences between terms.

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

This shows that the differences increase by 3 each time.

×2

×2

×2

×2

×2

×2

×2

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

For example, look at this sequence:

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

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.

÷4

÷4

÷4

÷4

÷4

÷4

÷4

We could also continue this sequence by multiplying by

each time.

1

4

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

512, 256, 64, 16, 4,

1,

0.25,

0.125, . . .

How did you work these out?

This sequence starts with 512 and decreases by dividing by 4 each time.

21+13

1+1

1+2

3+5

5+8

8+13

13+21

21+34

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

1, 1, 2, 3, 5, 8, 13,

21,

34,

55, . . .

How did you work these out?

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

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

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

- Sequences that increase or decrease in equal steps.
- These are calledlinearorarithmetic 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.

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

A number sequence starts as follows

1, 2, . . .

How many ways can you think of continuing the sequence?

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

A4 Sequences

Contents

A4.1 Introducing sequences

A4.2 Describing and continuing sequences

A4.3 Generating sequences

A4.4 Finding the nth term

A4.5 Sequences from practical contexts

No

Yes

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

START

This flow chart generates the sequence

3, 4.5, 6, 6.5, 9.

Write down 3.

Add on 1.5.

Write down the answer.

This sequence has only five terms.

Is the answer

more than 10?

It is finite.

STOP

No

Yes

START

Write down 5.

This flow chart generates the sequence

5, 2.9, 0.8, –1.3, –3.4.

Subtract 2.1.

Write down the answer.

Is the answer

less than -5?

STOP

No

Yes

START

Write down 200.

This flow chart generates the sequence

200, 100, 50, 25, 12.5, 6.25.

Divide by 2.

Write down the answer.

Is the answer

less than 4?

STOP

No

Yes

START

Write down 3 and 4.

This flow chart generates the sequence

3, 4, 7, 11, 18, 29, 47, 76.

Add together the two

previous numbers.

Write down the answer.

Is the answer

more than 100?

STOP

For example,

87, 84, 81, 78, . . .

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

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?

+1

+2

+3

+4

+5

+6

×2

×2

×2

×2

×2

×2

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

1

2

4

7

11

16

22

1

2

4

8

16

32

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.

+3

+3

+3

+3

+3

+3

This sequence continues by adding 3 each time.

1

4

7

10

13

16

19

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.

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,

5,

8,

11,

14,

17,

20,

23, . . .

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

0.4,

3.4,

6.4,

9.4,

12.4,

15.4,

18.4,

21.4, . . .

+2

+4

+6

+10

+12

+14

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

Term-to-term rule

5

Add consecutive even numbers starting with 2.

This gives us the sequence,

5

7

11

17

27

39

53

. . .

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

1st term

Term-to-term rule

10

Add 3

10,

13,

16,

19,

21

Subtract 5

100,

95,

90,

85,

80

100

3

Double

3,

6,

12,

24,

48

Multiply by 10

5,

50,

500,

5000,

50000

5

7

Subtract 2

7,

5,

3,

1,

–1

0.8

Add 0.1

0.8,

0.9,

1.0,

1.1,

1.2

Sometimes sequences are arranged in a table like this:

We can say that each term can be found by multiplying the position of the term by 3.

This is called a position-to-term rule.

For this sequence we can say that the nth term is 3n, where n is a term’s position in the sequence.

What is the 100th term in this sequence?

3 × 100 = 300

The position-to-term rule for a sequence is very useful because it allows us to work out any term in the sequence without having to work out any other terms.

We can use algebraic shorthand to do this.

We call the first term T(1), for Term number 1,

we call the second term T(2),

we call the third term T(3), . . .

we call the nth term T(n).

T(n) is called the the nth term or the general term.

For example, suppose the nth term of a sequence is 4n + 1.

We can write this rule as:

T(n) = 4n + 1

Find the first 5 terms.

T(1) =

4 ×1 + 1 =

5

T(2) =

4 ×2 + 1 =

9

T(3) =

4 ×3 + 1 =

13

T(4) =

4 ×4 + 1 =

17

T(5) =

4 ×5 + 1 =

21

The first 5 terms in the sequence are: 5, 9, 13, 17 and 21.

If the nth term of a sequence is 2n2 + 3.

We can write this rule as:

T(n) = 2n2 + 3

Find the first 4 terms.

T(1) =

2 ×12 + 3 =

5

T(2) =

2 ×22 + 3 =

11

T(3) =

2 ×32 + 3 =

21

T(4) =

2 ×42 + 3 =

35

The first 4 terms in the sequence are: 5, 11, 21, and 35.

This sequence is a quadratic sequence.

Which rule is best?

The term-to-term rule?

The position-to-term rule?

A4 Sequences

Contents

A4.1 Introducing sequences

A4.2 Describing and continuing sequences

A4.4 Finding the nth term

A4.3 Generating sequences

A4.5 Sequences from practical contexts

+5

+5

+5

+5

+5

+5

+5

× 5

× 5

× 5

× 5

× 5

× 5

All sequences of multiples can be generated by adding the same amount each time. They are linear sequences.

For example, the sequence of multiples of 5:

5, 10, 15, 20, 25, 30 35 40 …

can be found by adding 5 each time.

Compare the terms in the sequence of multiples of 5 to their position in the sequence:

2

10

3

15

4

20

5

25

n

Position

1

5

…

…

5n

Term

+3

+3

+3

+3

+3

+3

+3

×3

×3

×3

×3

×3

×3

The sequence of multiples of 3:

3, 6, 9, 12, 15, 18, 21, 24, …

can be found by adding 3 each time.

Compare the terms in the sequence of multiples of 3 to their position in the sequence:

2

6

3

9

4

12

5

15

n

Position

1

3

…

…

3n

Term

The nth term of a sequence of multiples is always dn, where d is the difference between consecutive terms.

The nth term of a sequence of multiples is always dn, where d is the difference between consecutive terms.

For example,

The nth term of 4, 8, 12, 16, 20, 24 … is

4n

The 10th term of this sequence is 4 × 10 = 40

The 25th term of this sequence is 4 × 25 = 100

The 47th term of this sequence is 4 × 47 = 188

+3

+3

+3

+3

+3

+3

+3

× 3

× 3

× 3

× 3

× 3

+ 1

+ 1

+ 1

× 3

+ 1

+ 1

+ 1

The terms in this sequence

4, 7, 10, 13, 16, 19, 22, 25 …

can be found by adding 3 each time.

Compare the terms in the sequence to the multiples of 3.

2

3

4

5

n

Position

1

…

Multiples of 3

3

6

9

12

15

3n

…

Term

4

7

10

13

16

3n + 1

Each term is one more than a multiple of 3.

+5

+5

+5

+5

+5

+5

+5

× 5

× 5

× 5

× 5

× 5

– 4

– 4

– 4

× 5

– 4

– 4

– 4

The terms in this sequence

1, 6, 11, 16, 21, 26, 31, 36 …

can be found by adding 5 each time.

Compare the terms in the sequence to the multiples of 5.

2

3

4

5

n

Position

1

…

Multiples of 5

5

10

15

20

25

5n

…

Term

1

6

11

16

21

5n– 4

Each term is four less than a multiple of 5.

–2

–2

–2

–2

–2

–2

–2

× –2

× –2

× –2

× –2

× –2

+ 7

+ 7

+ 7

× –2

+ 7

+ 7

+ 7

The terms in this sequence

5, 3, 1, –1, –3, –5, –7, –9 …

can be found by subtracting 2 each time.

Compare the terms in the sequence to the multiples of –2.

2

3

4

5

n

Position

1

…

Multiples of –2

–2

–4

–6

–8

–10

–2n

…

Term

5

3

1

–1

–3

7 – 2n

Each term is seven more than a multiple of –2.

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

The difference between any two consecutive terms in an arithmetic sequence is a constant number.

When we describe arithmetic sequences we call the difference between consecutive terms, d.

We call the first term in an arithmetic sequence, a.

For example, if an arithmetic sequence has a = 5 and d = -2,

We have the sequence:

5,

3,

1,

-1,

-3,

-5,

. . .

The rule for the nth term of any arithmetic sequence is of the form:

T(n) = an + b

a and b can be any number, including fractions and negative numbers.

For example,

Generates odd numbers starting at 3.

T(n) = 2n + 1

Generates even numbers starting at 6.

T(n) = 2n + 4

Generates even numbers starting at –2.

T(n) = 2n– 4

Generates multiples of 3 starting at 9.

T(n) = 3n + 6

Generates descending integers starting at 3.

T(n) = 4 –n

A4 Sequences

Contents

A4.1 Introducing sequences

A4.2 Describing and continuing sequences

A4.5 Sequences from practical contexts

A4.3 Generating sequences

A4.4 Finding the nth term

The following sequence of patterns is made from L-shaped tiles:

Number of

Tiles

4

8

12

16

The number of tiles in each pattern form a sequence.

How many tiles will be needed for the next pattern?

We add on four tiles each time.

This is a term-to-term rule.

A possible justification of this rule is that each shape has four ‘arms’ each increasing by one tile in the next arrangement.

The pattern give us multiples of 4:

1 lot of 4

2 lots of 4

3 lots of 4

4 lots of 4

The nth term is 4 ×n or 4n.

Justification: This follows because the 10th term would be 10 lots of 4.

Now, look at this pattern of blocks:

Number of

Blocks

4

7

10

13

How many blocks will there be in the next shape?

We add on 3 blocks each time.

This is the term-to term rule.

Justification: The shapes have three ‘arms’ each increasing by one block each time.

How many blocks will there be in the 100th arrangement?

We need a rule for the nth term.

Look at pattern again:

1st pattern

2nd pattern

3rd pattern

4th pattern

The nth pattern has 3n + 1 blocks in it.

Justification: The patterns have 3 ‘arms’ each increasing by one block each time. So the nth pattern has 3n blocks in the arms, plus one more in the centre.

So, how many blocks will there be in the 100th pattern?

Number of blocks in the nth pattern = 3n + 1

When n is 100,

Number of blocks =

(3 ×100) + 1 =

301

How many blocks will there be in:

a) Pattern 10?

(3 ×10) + 1 =

31

b) Pattern 25?

(3 ×25) + 1 =

76

c) Pattern 52?

(3 ×52) + 1 =

156

The number of blue tiles form the sequence 8, 13, 18, 32, . . .

Pattern

number

1

2

3

Number of

blue tiles

8

13

18

The rule for the nth term of this sequence is

T(n) = 5n + 3

Justification: Each time we add another yellow tile we add 5 blue tiles.

The +3 comes from the 3 tiles at the start of each pattern.