1. SEQUENCES

2. A sequence is a list of numbers or diagrams that are connected by a rule.15 17 19 21 …
and
are sequences.

3. The numbers in a sequence are called the terms of the sequence
The nth term is used to describe a general term in a sequence.
If the nth term = 3n + 5
1st term = 3 × = 8
2nd term = 3 × = 11
3rd term = 3 × = 14
4th term = 3 × = 17
The sequence is 8, 11, 14, 17, …
It is called a linear sequence because the differences between terms
are all the same.
The rule to find the next term (the term-to-term rule) is +3 or add 3.

4. 1 Find the nth term of the linear sequence 10, 17, 24, 31, …
Examples
1 Find the nth term of the linear sequence 10, 17, 24, 31, …
Term position
The rule is multiply by 7 and then add 3.
+ 7
+ 7
+ 7
nth term = 7n + 3

5. 2 Find the nth term of the linear sequence 2, 5, 8, 11, …
Examples
2 Find the nth term of the linear sequence 2, 5, 8, 11, …
Term position
The rule is multiply by 3 and then take 1.
+ 3
+ 3
+ 3
nth term = 3n - 1

6. 3 Find the nth term of the linear sequence 9, 13, 17, 21, …
Examples
3 Find the nth term of the linear sequence 9, 13, 17, 21, …
Term position
The rule is multiply by 4 and then add 5.
+ 4
+ 4
+ 4
nth term = 4n + 5

7. 4 Find the nth term of the linear sequence 29, 22, 15, 8, …
Examples
4 Find the nth term of the linear sequence 29, 22, 15, 8, …
Term position
The rule is multiply by −7 and then add 36. −7 −7 −7
nth term = 36 − 7n

8. b How many circles are in diagram 8?
Examples 5
diagram 1
diagram 2
diagram 3
diagram 4
a Complete the table.
Diagram 1 2 3 4
Number of circles
× 4 4 8 12 16
+ 4
+ 4
+ 4
Answer = 8 × 4 = 32
b How many circles are in diagram 8?
c How many circles are in diagram n?
Answer = 8 × n = 8n

9. b How many sticks are in diagram 27? Answer = 27 × 3 + 1 = 82
Examples 6
diagram 1
diagram 2
diagram 3
a Complete the table.
Diagram 1 2 3 4
Number of sticks
× 3 then +1 4 7 10 13
+ 3
+ 3
+ 3
b How many sticks are in diagram 27?
Answer = 27 × = 82
c How many sticks are in diagram n?
Answer = 3n + 1

10. Examples
7 nth term = 5n Find the term in the sequence that has a value of 122.
add 3 to both sides
divide both sides by 5
Answer: the 25th term

11. Examples
8 nth term = 6n Find the term in the sequence that has a value of 303.
take 15 from both sides
divide both sides by 6
Answer: the 48th term

12. Examples
9 nth term = n. Find the term in the sequence that has a value of -184.
take 50 from both sides
divide both sides by −6
Answer: the 39th term

13. Non linear sequences
The sequence 5, 7, 10, 14, … is called a non-linear sequence because the difference between the terms is not the same.
+ 2
+ 3
+ 4
You should be able to recognise and use the following non-linear sequences.
Square numbers 1 4 9 16
nth term = n2

14. Cube numbers 1 8 27 64
nth term = n3
Powers
For example 2, 4, 8, 16, … nth term = 2n
3, 9, 27, 81, … nth term = 3n
4, 16, 64, 256, … nth term = 4n

15. Examples
1 Find the nth term of the sequence 1, 3, 9, 27, …
Term position
The rule involves powers of 3.
(= 30)
(= 31)
(= 32)
(= 33)
subtract 1 from the term position to find the power
nth term =

16. Examples
2 Find the nth term of the sequence 4, 7, 12, 19, …
Term position
The rule involves square numbers.
(= )
(= )
(= )
(= )
square the term position and add 3
nth term =

17. Examples
3 Find the nth term of the sequence 0, 7, 26, 63, …
Term position
The rule involves cube numbers.
(= 13 − 1)
(= 23 − 1)
(= 33 − 1)
(= 43 − 1)
cube the term position and subtract 1
nth term =

18. Examples
4 The table shows the first four terms in three sequences A, B and C.
Term 1
Term 2
Term 3
Term 4
Sequence A 3 6 11 18
Sequence B 9 27 81
Sequence C 16 63
a Find the nth term of sequence A.
b Find the nth term of sequence B.
c Find the nth term of sequence C.
You need to notice that: sequence C = sequence B − sequence A

19. b How many circles are in diagram 20? Answer =
Examples 5
diagram 1
diagram 2
diagram 3
diagram 4
a Complete the table.
Diagram 1 2 3 4 5
Number of circles
b How many circles are in diagram 20?
Answer =
c How many circles are in diagram n?
Answer =

20. The diagram shows the growth of a plant over three years.
Examples 6
Year 1
Year 2
Year 3
The diagram shows the growth of a plant over three years.
Each year a flower is replaced by three stems and three flowers.
a Complete the table.
Year 1 2 3 4 5
Number of flowers 9
Number of stems 13 27 81 40
121

21. b How many flowers are there in year n?
Examples
Year 1
Year 2
Year 3
b How many flowers are there in year n?
Using the answers from part a you can see a pattern.
Year 1 2 3 4 5
Number of flowers
1 = 30
3 = 31
9 = 32
27= 33
81 = 34
So, number of flowers in year n =

22. c How many stems are there in year n?
Examples
Year 1
Year 2
Year 3
c How many stems are there in year n?
Using the answers from part a you can see a pattern.
Year 1 2 3 4 5
Number of stems
So, number of stems in year n =

23. Further non-linear sequences
Example
1 The nth term of the sequence 8, 16, 26, 38, … is given by the formula
nth term = n2 + bn + c
Find the values of b and c.
The first term is 8, so substitute n = 1 into the formula
The second term is 16, so substitute n = 2 into the formula
Solving and simultaneously gives b = 5 and c = 2.

24. If you are asked to find the values of a, b and c when the
nth term = an2 + bn + c
then the problem is more complicated.
One method is to look at the differences in the terms. For example
First difference
+ 9
+ 13
+ 17
+ 21
Second difference
+ 4
+ 4
+ 4
The second differences are all the same.
This means that it is a quadratic sequence
A second difference of 2  the nth term will contain n2.
A second difference of 4  the nth term will contain 2n2.
A second difference of 6  the nth term will contain 3n2.
So divide the second difference by 2 to find the coefficient of n2.

25. How many sticks are needed to make
Examples 2
diagram 1
diagram 2
diagram 3
3 sticks
9 sticks
18 sticks
How many sticks are needed to make
a diagram 4 b diagram 5 c diagram n?
a diagram 4 = = 30 sticks
b diagram 5 = = 45 sticks

26. solving simultaneously
Examples 2
diagram 1
diagram 2
diagram 3
3 sticks
9 sticks
18 sticks c
second difference is 3,
so a = 3 ÷ 2 = 1.5
+ 6
+ 9
+ 12
+ 15
+ 3
+ 3
+ 3
and
simplifying
and
solving simultaneously or