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First teaching 2021

Last exams 2024

Sequences (CIE IGCSE Maths: Core)

Revision Note

Test Yourself

Author

Amber

Expertise

Maths

Introduction to Sequences

What are sequences?

A sequence is an ordered set of (usually) numbers
Each number in a sequence is called a term
The location of a term within a sequence is called its position
- The letter n is often used for (an unknown) position
Subscript notation is used to talk about a particular term
- a₁ would be the first term in a sequence
- a₇ would be the seventh term
- a_n would be the n^th term

SeqBas Notes fig1, downloadable IGCSE & GCSE Maths revision notes

What is a position-to-term rule?

A position-to-term rule gives the n^th term of a sequence in terms of n
- This is a very powerful piece of mathematics
- With a position-to-term rule the 100^th term of a sequence can be found without having to know or work out the first 99 terms!

SeqBas Notes fig2, downloadable IGCSE & GCSE Maths revision notes

What is a term-to-term rule?

A term-to-term rule gives the (n+1)^th term in terms of the n^th term
- ie a_n+1 is given in terms of a_n
- If a term is known, the next one can be worked out

SeqBas Notes fig3, downloadable IGCSE & GCSE Maths revision notes

How do I use the position-to-term and term-to-term rules?

These can be used to generate a sequence
If the term to term rule is known, a sequence can be continued by using the rule to find each term from the next
- For example, if the first term of a sequence is 2 and the term to term rule is 'add 3' then the first 5 terms are 2, 5, 8, 11, 14
If the position to term rule is known, any term can be found in the sequence by substituting the value of the position (usually n) into the rule
- For example, if the position to term rule is 2n -1, the 50th term is 2(50) - 1 = 99

SeqBas Notes fig4, downloadable IGCSE & GCSE Maths revision notes

How do I determine if a number appears within a sequence?

If you know the term-to-term rule you could continue the sequence until you get to the number or until you pass the number
If you know the position-to-term rule then you can form an equation
- Make the formula equal to the number
- Solve for n
- If n is a natural number then the number appears in the sequence
- If n is a decimal or negative then the number does not appear in the sequence

Exam Tip

Write the position numbers above (or below) each term in a sequence
- This will make it much easier to recognise and spot common types of sequence
Always start any sequences problem by looking for the term to term rule, if it can be easily determined it will usually be helpful

Worked example

For each of the following sequences, find the first 5 terms and the 20th term.

Position to term rule = 3n + 2

ii)

Position to term rule = 20 - 2n

iii)

First term = 5 and term to term rule = 'add 1'

Find each term individually by substituting the value of each position into the position to term rule.

1st term → 3(1) + 2 = 5
2nd term → 3(2) + 2 = 8
3rd term → 3(3) + 2 = 11
4th term → 3(4) + 2 = 14
5th term → 3(5) + 2 = 17

20th term → 3(20) + 2 = 62

5, 8, 11, 14, 17

ii)

Find each term individually by substituting the value of each position into the position to term rule.

1st term → 20 - 2(1) = 18
2nd term → 20 - 2(2) = 16
3rd term → 20 - 2(3) = 14
4th term → 20 - 2(4) = 12
5th term → 20 - 2(5) = 10

20th term → 20 - 2(20) = -20

18, 16, 14, 12, 10

-20

iii)

Find the first 5 terms in turn by adding 1 to the term before it.

1st term → 5
2nd term → 5 + 1 = 6
3rd term → 6 + 1 = 7
4th term → 7 + 1 = 8
5th term → 8 + 1 = 9

The 20th term can be found by either continuing to find each term in the sequence, up to the 20th term, or by finding the position to term rule.

Each term is 4 more than its position in the sequence, so the position to term rule is n + 4, where n is the position in the sequence.

20th term → 20 + 4 = 24

5, 6, 7, 8, 9

Continuing Sequences

What types of sequences are there?

Linear, quadratic and cubic sequences are particular types of sequence covered their own notes
Other sequences include geometric and Fibonacci sequences
Another common type of sequence in exam questions, is fractions with combinations of the above
- Look for anything that makes the position-to-term and/or the term-to-term rule easy to spot
To be able to continue sequences, you need to be able to spot what type of sequence you are looking at

SeqOth Notes fig2, downloadable IGCSE & GCSE Maths revision notes

What is a Fibonacci sequence?

THE Fibonacci sequence is 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, ...
The sequence starts with the first two terms as 1
Each subsequent term is the sum of the previous two
- ie The term-to-term rule is a_n+2 = a_n+1 + a_n
- Notice that two terms are needed to start a Fibonacci sequence
Any sequence that has the term-to-term rule of adding the previous two terms is called a Fibonacci sequence but the first two terms will not both be 1
Fibonacci sequences occur a lot in nature such as the number of petals of flowers

SeqOth Notes fig4, downloadable IGCSE & GCSE Maths revision notes

How do I continue a given sequence?

You may be given the first few terms of a sequence and be asked to find the next few terms
- The key is to recognise the type of sequence and use the term to term rule to continue the sequence
CASE 1: Look first to see if the sequence has a common difference
- If it does then it is a linear sequence
  - Find the next terms by adding or subtracting that common difference to each term in turn
  - This is an example of using the term to term rule
CASE 2: If the sequence does not have a common difference then look to see if it has a common second or third difference
- If it has a common second difference then it is a quadratic sequence
- If it has a common third difference then it is a cubic sequence
  - Find the next term by first finding the next difference and then adding this to the sequence
CASE 3: If there is no second or third common difference then look to see if it has a common multiplier (ratio)
- If it does then it is a geometric sequence
  - Find the next term by multiplying the last term by the common multiplier
- Look to see if there is a common value you can multiply each term by to get to the next term
CASE 4: Some sequences may not fit into any of these categories, in this case you should look to see if there is a pattern in the difference which could help you to find the next term
- For example the differences may form a geometric sequence which can be used to find the next difference and hence, the next value in the sequence
You may also be asked to fill in some gaps within a sequence
- In this case you will need to use the information you have to determine the type of sequence and to fill in the gaps

Worked example

Find the term to term rule in the following sequence and hence, find the next term.

18, 25, 32, 39, 46

Look for a common difference in the sequence.

18 → +7 = 25

25 → +7 = 32

32 → +7 = 39

39 → +7 = 46

The common difference is +7 so this is a linear sequence with term to term rule of 'add 7'.

Find the next term by adding 7 to the last given term

46 → +7 = 53

Term to term rule = add 7

Next term = 53

Find the next term in the sequence

16, 18, 22, 30, 46

Look for a common difference in the sequence.

16 → +2 = 18
18 → +4 = 22
22 → +8 = 30
30 → +16 = 46

There is no common difference so this is not a linear sequence.
There is no common second difference to this is not a quadratic sequence.
This is also not a geometric sequence as there is not a number you can multiply a term by to get to the next term.

To find the next term you will need to look at the pattern in the differences, 2, 4, 8, 16. This is a geometric sequence with term to term rule of 'multiply by 2', so the next term in this sequence can be found.

The differences are doubling each time, so continuing this pattern, the next difference will be 2 × 16 = + 32

46 → +32 = 78

Next term = 78

The 1st and 3rd terms in a Fibonacci sequence are 2 and 7 respectively.

Find the 2nd and 4th terms of the sequence.

Write what you know about the sequence.

$n$	$1$	$2$	$3$	$4$
$a_{n}$	$2$	$x$	$7$	$x + 7$

Form an equation by adding the first two terms and setting them equal to the third term, 7.

$2 + x = 7$

Solve the equation to find the value of the second term.

$x = 7 - 2 x = 5$

Add the second and third terms together to find the value of the fourth term.

$a_{4} = x + 7 = 5 + 7 = 12$

2nd term = 5, 4th term = 12

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