A21 sequences and rules

© Boardworks Ltd 20091 of 15

A21 Sequences and rules

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© Boardworks Ltd 20092 of 25 © Boardworks Ltd 20092 of 15

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.

Continuing sequences


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.

Continuing sequences

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.

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.

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,


2nd1st

3Term

Sometimes sequences are arranged in a table like this:

nth…6th5th4th3rdPosition

Sequences from position-to-term rules

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

3n…18151296


Sequences from position-to-term rules


Writing sequences from position-to-term rules

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), ...

and 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:

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.

T(n) = 2n2 + 3


Sequence generator – linear sequences


Sequence generator – non-linear sequences


Which rule is best?

Sequences and rules

The term-to-term rule?

The position-to-term rule?

A21 sequences and rules

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