GCSE Maths · Higher · Chapter 4: Number and sequences · Lesson 2

Special sequences and rules from patterns

Grades 4–7About 50 minutesNo calculator36 marks

Hints online, answers free with an account:
mathswariz.uk/worksheets/gcse-special-sequences-and-rules-from-patterns/

NameClassDate

Do Now

Grades 4–6

The first question is what today's lesson needs. The others bring back earlier topics.

A1
Needed today
Work out $196$ .
[1 mark]
Answer
A2
Last lesson
Find the $n$ th term of $2, 6, 10, 14, \dots$
[1 mark]
Answer
A3
From chapter 2
A price rises from £250 to £312.50. Find the percentage increase.
[1 mark]
Answer
A4
Key Stage 3
Work out $152 \div 8$ .
[1 mark]
Answer

Grades 5–6

Your teacher works through each example with you. Then try the one beside it.

Example 1

Find the $n$ th term of $10$ , $30$ , $90$ , $270$ , ... and work out the $6$ th term.

Hint Each term is the one before times

3

B1
Find the $n$ th term of $13$ , $26$ , $52$ , $104$ , ... and work out the $7$ th term.
[3 marks]
Answer $\dots\dots$ and $\dots\dots$

Example 2

A sequence starts $5$ , $p$ , and each term after that is the sum of the two before. The $4$ th term is $29$ . Find $p$ and the $5$ th term.

Hint Write the

3

rd and

4

th terms in terms of

p

B2
A sequence starts $3$ , $b$ , and each term after that is the sum of the two before. The $4$ th term is $23$ . Find $b$ and the $5$ th term.
[3 marks]
Answer $b = \dots\dots$ and $\dots\dots$

Grades 5–6

C3
A geometric sequence has first term $640$ and common ratio $\frac{1}{2}$ . Work out the $6$ th term.
[2 marks]
Answer
C4
Which term of the sequence with $n$ th term $n^{2} + 4$ is equal to $200$ ?
[2 marks]
Answer
C5
Work out the first term of the sequence $5$ , $15$ , $45$ , $135$ , ... that is greater than $2000$ .
[2 marks]
Answer
C6
In a geometric sequence the $2$ nd term is $6$ and the $4$ th term is $54$ . Every term is positive. Work out the $6$ th term.
[2 marks]
Answer

Grades 6–7

D1
Here are the first four terms of a sequence: $1$ , $8$ , $27$ , $64$ . Mia says, "The $10$ th term is $30$ , because it is $3 \times 10$ ."
Explain why Mia is wrong, and write down the correct $10$ th term.
[2 marks]

D2
The diagram shows the first three patterns in a sequence. Pattern $n$ is $3$ rows of $n$ squares made from matchsticks.
(a)
Find an expression, in terms of $n$ , for the number of matchsticks in pattern $n$ .
[2 marks]
Answer
(b)
Show that no pattern uses exactly $100$ matchsticks.
[2 marks]
(c)
One pattern uses $213$ matchsticks. How many squares does it have? You must show your working.
[3 marks]
Answer
Total for D2: 7 marks
D3
The first three terms of a geometric sequence are $x$ , $x + 12$ and $x + 30$ . Work out the value of $x$ and the $4$ th term of the sequence. You must show your working.
[4 marks]
Answer $x = \dots\dots$ fourth term $\dots\dots$

Grade 8 and beyond

No route is given. The hints are at the end of the sheet. Use one at a time.

Colour one circle on each line.

	Not yet	Nearly	Yes
I can recognise square, cube and triangular numbers and continue them.
I can find and use the $n$ th term of a geometric sequence and a sequence that adds the two before.
I can find a rule from a growing pattern and use it.

The worked answers and the full mark scheme are free with a Mathswariz account. Teachers and students can both sign up.