IGCSEMaths: ExtendedCIETopic Questions2. Algebra & Graphs2.11 Sequences

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First teaching 2021

Last exams 2024

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Sequences (CIE IGCSE Maths: Extended)

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1a
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Find the nth term of each sequence.

4 Ā  Ā  Ā  Ā 2 Ā  Ā  Ā  Ā 0 Ā  Ā  Ā  Ā -2 Ā  Ā  Ā  Ā -4 Ā  Ā  Ā  Ā ...



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1b
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1 Ā  Ā  Ā  Ā 7 Ā  Ā  Ā  Ā 17 Ā  Ā  Ā  Ā 31 Ā  Ā  Ā  Ā 49 Ā  Ā  Ā  Ā ...

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2
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Complete the table for the 5th term and the nth term of each sequence.

1st term 2nd term 3rd term 4th term 5th term Ā  nth term
9 5 1 -3 Ā  Ā  Ā 
4 9 16 25 Ā  Ā  Ā 
1 8 27 64 Ā  Ā  Ā 
8 16 32 64 Ā  Ā  Ā 

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3
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0,Ā  Ā 1,Ā  Ā 1,Ā  Ā 2,Ā  Ā 3,Ā  Ā 5,Ā  Ā 8,Ā  Ā 13,Ā  Ā 21Ā  ,Ā  Ā ...

This sequence is a Fibonacci sequence.
After the first two terms, the rule to find the next term is ā€œadd the two previous termsā€.
For example, 5 + 8 = 13.

Use this rule to complete each of the following Fibonacci sequences.

space space space space 2 space space space space space space space space space space space space space space space space 4 space space space space space space space space space space space space space space........ space space space space space space space space space space........ space space space space space space space space space space........ space space space space 1 space space space space space space space space space space space space........ space space space space space space space space space space........ space space space space space space space space space space........ space space space space space space space space space space space space space 11 ........ space space space space space space space space space space minus 1 space space space space space space space space space space space space........ space space space space space space space space space space........ space space space space space space space space space space space space space space 1

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4
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1 third comma space space space space space 3 over 4 comma space space space space space 4 over 7 comma space space space space space 7 over 11 comma space space space space space 11 over 18 comma space space space space space...

i)begin mathsize 16px style table row blank row blank end table end style
One term of this sequence is p over q.

Find, in terms of p and q, the next term in this sequence.

[1]

ii)
Find the 6th term of this sequence.

[1]

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5
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Find the nth term of each sequence.

i)
negative 1 comma space space space space space space space minus 3 comma space space space space space space space minus 5 comma space space space space space space space minus 7 comma space space space space space space space minus 9 comma space space space space space space space...
Ā 
Ā 

[2]

ii)
2 comma space space space space space space space 9 comma space space space space space space space 28 comma space space space space space space space 65 comma space space space space space space space 126 comma space space space space space space space...
Ā 
Ā 

[2]

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6
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Find the next term and the nth term of this sequence.

3 over 5 comma space space space space space space space space 4 over 7 comma space space space space space space space space 5 over 9 comma space space space space space space space space 6 over 11 comma space space space space space space space space 7 over 13 comma space space space space space space space...

Next term = .................................................Ā Ā 

nth term = .................................................Ā 

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1a
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Marco is making patterns with grey and white circular mats.
cie-igcse-2018-may-jun-1-p4-q12
The patterns form a sequence.
Marco makes a table to show some information about the patterns.

Pattern number 1 2 3 4 5
Number of grey mats 6 9 12 15 Ā 
Total number of mats 6 10 15 21 Ā 

Complete the table for Pattern 5.

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1b
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Find an expression, in terms of n, for the number of grey mats in Pattern n.

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1c
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Marco makes a pattern with 24 grey mats.

Find the total number of mats in this pattern.

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2a
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19, 15, 11, 7, ....

Write down the next two terms of the sequence.

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2b
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Find the nth term of this sequence.
Ā 

[2]

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2c
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Find the value of n when the nth term is -65.
Ā 

n = ..............................................Ā 

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3
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A sequence has nth termĀ 2 n squared plus 5 n minus 15.
Ā 
Find the difference between the 4th term and the 5th term of this sequence.

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4
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TheĀ nth term of a sequence is space 4 n squared plus n plus 3.

i)

Find the 2nd term.
Ā 

[1]

ii)

Find the value ofĀ n when the nth term is 498.
Ā 

n equals.................................................. [3]

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5a
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Here are the first six terms of a Fibonacci sequence.

Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā 1 Ā Ā 1 Ā Ā 2 Ā Ā 3 Ā  Ā  Ā Ā  Ā 5Ā Ā Ā  8

The rule to continue a Fibonacci sequence is,

Ā  Ā Ā Ā Ā Ā Ā Ā the next term in the sequence is the sum of the two previous terms.

Find the 9th term of this sequence.

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5b
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The first three terms of a different Fibonacci sequence are

a space space space space space space space b space space space space space space space a plus b

Show that the 6th term of this sequence isĀ 3 a space plus space 5 b

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5c
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Given that the 3rd term is 7 and the 6th term is 29,

find the value of a and the value of b.

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1a
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cie-igcse-2020-oct-nov-p4-tz1-q7

These are the first four diagrams of a sequence.
The diagrams are made from white dots and black dots.

Complete the table for Diagram 5 and Diagram 6.

Diagram 1 2 3 4 5 6
Number of white dots 1 4 9 16 Ā  Ā 
Number of black dots 0 1 3 6 Ā  Ā 
Total number of dots 1 5 12 22 Ā  Ā 

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1b
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Write an expression, in terms of n, for the number of white dots in Diagram n.

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1c
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The expression for the total number of dots in DiagramĀ n isĀ  1 half open parentheses 3 n squared space minus space n close parentheses.

i)
Find the total number of dots in Diagram 8.
Ā  Ā 
[1]
ii)
Find an expression for the number of black dots in Diagram n.
Give your answer in its simplest form.
Ā  Ā 
[2]

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1d
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T spaceis the total number of dots used to make all of the firstĀ  n diagrams.

T thin space equals space a n to the power of 3 space end exponent plus space b n squaredĀ 

Find the value ofĀ a and the value of b.
You must show all your working.


a space equals ................................................Ā 

b space equals space................................................Ā 

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2a
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Sequence 1st term 2nd term 3rd term 4th term 5th term Ā  nth term
A 13 9 5 1 Ā  Ā  Ā 
B 0 7 26 63 Ā  Ā  Ā 
C 7 over 8 8 over 16 9 over 32 10 over 64 Ā  Ā  Ā 

Complete the table for the three sequences.

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2b
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One term in Sequence C isĀ p over q.
Write down the next term in Sequence C in terms of space p spaceand q.

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3
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The table shows the first four terms in sequences A, B, and C.

Sequence 1st term 2nd term 3rd term 4th term 5th term Ā  nth term
A 4 9 14 19 Ā  Ā  Ā 
B 3 10 29 66 Ā  Ā  Ā 
C 1 4 16 64 Ā  Ā  Ā 


Complete the table.

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4a
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cie-igcse-2019-may-jun-p4-tz2-q11a

The sequence of diagrams above is made up of small lines and dots.

Complete the table.

Ā  Diagram 1 Diagram 2 Diagram 3 Diagram 4 Diagram 5 Diagram 6
Number of small lines 4 10 18 28 Ā  Ā 
Number of dots 4 8 13 19 Ā  Ā 

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4b
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For Diagram n find an expression, in terms of n, for the number of small lines.Ā 

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4c
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Diagram r has 10 300 small lines.
Ā 
Find the value ofĀ r.
Ā 

r = ..............................................Ā 

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4d
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The number of dots in Diagram n isĀ a n squared plus b n plus 1.
Ā 
Find the value of a and the value of b.
Ā 

a = ..............................................Ā 
b = ..............................................Ā 

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5a
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The table shows the first five terms of sequence A and sequence B.

Term 1 2 3 4 5 6
SequenceĀ A 7 13 23 37 55 Ā 
SequenceĀ B 1 3 9 27 81 Ā 

Complete the table for the 6th term of each sequence.

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5b
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i)
Find the nth term of sequence A.
Ā 
[2]
ii)
Find the nth term of sequenceĀ B.
Ā 
[2]

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6a
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The table shows the first five terms of sequences A, B and C.

Sequence 1st term 2nd term 3rd term 4th term 5th term 6th term
A 0 1 4 9 16 Ā 
B 4 5 6 7 8 Ā 
C -4 -4 -2 2 8 Ā 

Complete the table.

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6b
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Find an expression for the nth term of

i)

sequenceĀ A,
Ā 

[2]

ii)

sequenceĀ B.
Ā 

[1]

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6c
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Find the value ofĀ n when the nth term of sequence A is 576.
Ā 

n equals...............................................

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6d
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i)

Find an expression for the nth term of sequence C.
Give your answer in its simplest form.
Ā 

[3]

ii)

Find the value of the 30th term of sequence C.
Ā 

[2]

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