GCSE Maths · Higher · Chapter 4: Number and sequences

Linear, special and quadratic sequences

Grades 4–9About 80 minutesNo calculator87 marks

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Before you start

Grades 4–5

You will need these in every question below.

A1
Write down the next two terms of the sequence 12, 21, 30, 39, …
[1 mark]
Answer

A2
Write down the term-to-term rule of the sequence 40, 34, 28, 22, …
[1 mark]
Answer
A3
Work out the value of $3 n - 2$ when $n = 9$ .
[1 mark]
Answer
A4
Write down the first three square numbers that are greater than 50.
[1 mark]
Answer
A5
The first term of a sequence is 4. The term-to-term rule is "multiply by 3". Work out the fourth term.
[1 mark]
Answer

Examples, then your turn

Grades 5–6

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

Example 1

Find the nth term of the sequence 2, 11, 20, 29, …

Hint The terms go up by 9, so the nth term starts

9 n

. Compare

9 n

with the sequence to find what to add or subtract.

B1
Find the nth term of the sequence 6, 14, 22, 30, …
[2 marks]
Answer

Example 2

Is 97 a term of the sequence with nth term $3 n + 4$ ?

Hint Set the nth term equal to 97 and solve. The term number has to be a whole number.

B2
Is 123 a term of the sequence with nth term $5 n - 3$ ? Give a reason.
[2 marks]
Answer

Example 3

Find the nth term of the sequence 4, 9, 18, 31, 48, …

Hint Find the first differences, then the second differences. Half the second difference is the number in front of

n^{2}

. Take that away from each term and find the nth term of what is left.

B3
Find the nth term of the sequence 4, 12, 26, 46, 72, …
[3 marks]
Answer

Example 4

Pattern 1 uses 6 tiles, pattern 2 uses 10 tiles and pattern 3 uses 14 tiles. The patterns carry on in the same way. How many tiles are in pattern 40?

Hint Each pattern uses 4 more tiles than the one before. Find the nth term first, then put

n = 40

into it.

B4
Pattern 1 uses 7 matches, pattern 2 uses 12 matches and pattern 3 uses 17 matches. The patterns carry on in the same way. How many matches are in pattern 30?
[2 marks]
Answer

Practice

Grades 5–7

Each question asks for something different. Show your working.

C1
Here is a sequence: 41, 37, 33, 29, …
(a)
Find the nth term.
[2 marks]
Answer
(b)
Work out the first term of the sequence that is negative.
[2 marks]
Answer
Total for C1: 4 marks

C2
In a sequence, each term after the second is the sum of the two terms before it. The first term is 2 and the second term is 5. Work out the seventh term.
[2 marks]
Answer
C3
Here is a sequence: 1, 3, 6, 10, 15, …
Write down the next term, and the name of this sequence.
[2 marks]
Answer
C4
The diagram shows the first three patterns in a sequence of dots. Pattern $n$ has $n$ rows of dots.
(a)
How many dots are in pattern 4?
[1 mark]
Answer
(b)
Write down an expression for the number of dots in pattern $n$ .
[2 marks]
Answer
(c)
Which pattern has 180 dots?
[2 marks]
Answer
Total for C4: 5 marks
C5
Here is a geometric sequence: 81, 54, 36, 24, …
Write down the common ratio and work out the next term.
[2 marks]
Answer
C6
Find the nth term of the sequence 6, 16, 32, 54, 82, …
[3 marks]
Answer
C7
The nth term of a sequence is $n^{2} - 3 n + 5$ . Which term of the sequence is equal to 59?
[3 marks]
Answer
C8
Sequence A has nth term $3 n + 2$ . Sequence B has nth term $62 - 3 n$ . For one value of $n$ the two sequences have the same term. Find that value of $n$ and the term.
[3 marks]
Answer
C9
Find the nth term of the sequence 0.5, 0.8, 1.1, 1.4, …
[2 marks]
Answer

Exam-style questions

Grades 6–9

Answer every question in the space given. Show your working: most of the marks are for method.

D1
Jo says, "The nth term of the sequence 7, 16, 25, 34, … is $n + 9$ , because you add 9 each time."
Explain why Jo is wrong, and write down the correct nth term.
[3 marks]
Answer

D2
Here is a sequence: 7, 13, 19, 25, …
Is 200 a term of this sequence? You must show your working.
[3 marks]
Answer
D3
Here is a quadratic sequence: 3, 12, 25, 42, 63, …
(a)
Find the nth term.
[3 marks]
Answer
(b)
Work out the 20th term.
[2 marks]
Answer
Total for D3: 5 marks
D4
The diagram shows the first three patterns in a sequence made from square tiles. The patterns carry on in the same way.
Gemma has 300 tiles. She makes the largest pattern in the sequence that she can.
How many tiles does she have left over? You must show your working.
[6 marks]
Answer
D5
In a sequence, each term after the second is the sum of the two terms before it. The third term is 15 and the fifth term is 41.
Find the first term and the second term.
[4 marks]
Answer
D6
Find the nth term of the sequence $\frac{3}{5}, \frac{5}{8}, \frac{7}{11}, \frac{9}{14}, \dots$
[3 marks]
Answer
D7
The nth term of a sequence is $85 - 6 n$ .
(a)
Show that 31 is a term of the sequence.
[2 marks]
(b)
How many terms of the sequence are positive?
[2 marks]
Answer
Total for D7: 4 marks
D8
The 3rd term of a linear sequence is 17 and the 7th term is 41.
Find the nth term, and the first term of the sequence that is greater than 100.
[4 marks]
Answer
D9
The nth term of a sequence is $a n^{2} + bn$ . The first term is 1 and the second term is 8.
Find the values of $a$ and $b$ , and work out the third term.
[3 marks]
Answer
D10
The nth triangular number is $\frac{n ( n + 1 )}{2}$ .
Show that the sum of any two triangular numbers next to each other in the sequence is a square number.
[3 marks]

Extension

Grade 9 and beyond

No route is given. Spend five minutes on a problem before you look at a hint. The hints are on the last page of the worksheet, and you can use one at a time.

E1
Find the nth term of the sequence 1, 7, 19, 37, 61, …
[3 marks]
Hint 1 · What to try
Work out the first differences and then the second differences.
Hint 2 · The first line
The second difference is 6, so the nth term starts $3 n^{2}$ . Take $3 n^{2}$ away from each term.
Hint 3 · The full method
$1 - 3 = - 2$ , $7 - 12 = - 5$ , $19 - 27 = - 8$ : this is $1 - 3 n$ . So the nth term is $3 n^{2} - 3 n + 1$ .
Answer

E2
The first 100 terms of the sequence with nth term $4 n + 3$ are written down.
How many of them are square numbers?
[3 marks]
Hint 1 · What to try
Write down the first few terms and the first few square numbers. What do you notice when you divide them by 4?
Hint 2 · The first line
Every term $4 n + 3$ leaves a remainder of 3 when you divide it by 4. An even square, $(2 k)^{2} = 4 k^{2}$ , leaves remainder 0.
Hint 3 · The full method
An odd square is $(2 k + 1)^{2} = 4 k^{2} + 4 k + 1$ , which leaves remainder 1. No square leaves remainder 3, so none of the terms is a square number: the answer is 0.
Answer
E3
In a sequence, each term after the second is the sum of the two terms before it. The first term is 2 and the sixth term is 46.
Find the second term.
[3 marks]
Hint 1 · What to try
Call the second term $b$ . Write the third, fourth, fifth and sixth terms in terms of $b$ .
Hint 2 · The first line
The terms are $2$ , $b$ , $2 + b$ , $2 + 2 b$ , $4 + 3 b$ , $6 + 5 b$ .
Hint 3 · The full method
$6 + 5 b = 46$ , so $5 b = 40$ and the second term is $8$ .
Answer

How did it go?

Colour one circle on each line.

	Not yet	Nearly	Yes
I can find the nth term of a linear sequence and use it to test whether a number is a term.
I can recognise square, cube, triangular and Fibonacci-type sequences and write a rule for a growing pattern.
I can find the nth term of a quadratic sequence using second differences.

Answers and mark scheme

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