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Recognising Quadratic and Geometric Sequences

In this worksheet, students will practise recognising and continuing geometric and quadratic sequences.

'Recognising Quadratic and Geometric Sequences' worksheet

Key stage:  KS 4

GCSE Subjects:   Maths

GCSE Boards:   AQA, Eduqas, OCR, Pearson Edexcel

Curriculum topic:   Algebra

Curriculum subtopic:   Sequences

Difficulty level:  

Worksheet Overview

QUESTION 1 of 10

There are a number of different types of sequences, we have already looked at linear sequences where the number goes up or down by the same each time.

We need to now be able to work with two more types of sequence.

Geometric: A seqeunce that has a common ratio. This means we multiply by the same amount each time.

Quadratic: A quadratic sequnce has a common second difference. This means that the difference between the differences will be the same.

 

Example 1: Describe the rule for, and give the next two terms of the sequence 3, 6, 12, 24...

We should notice that the sequence is doubling each time, this means the rule (the common ratio) is x 2

If we then continue the sequence, the next two terms are 48 and 96.

 

Example 2: Find the next term in the sequence 3, 6, 11, 18, 27

We first have to identify the sequence type here. It isn't going up by the same (so it can't be linear), it isn't multiplying each time, so it cant be a geometric sequence. This leaves a quadratic.

Step 1: Find the difference between the terms.

The difference between the first ans second terms is 3

The difference between the second and third terms is 5

The difference between the third and fourth terms is 7

The difference between the fourth and fifth terms is 9

Do you see how the difference is going up by 2 each time? this means...

The difference between the fifth and sixth terms is 11

 

This means the next term in our sequence 3, 6, 11, 18, 27 will be 38

Complete the following sentence...

Match these sequences with the correct common ratio

Column A

Column B

1, 2, 4, 8
x 0.1
100, 10, 1
x 2
5, 15, 75
x 1
10, 10, 10, 10
x 3

Complete the following sentence

Column A

Column B

1, 2, 4, 8
x 0.1
100, 10, 1
x 2
5, 15, 75
x 1
10, 10, 10, 10
x 3

Match the sequence to its next value.

Column A

Column B

1, 4, 9, 16...
50
2, 8, 18, 32...
25
0, 3, 8, 15...
24

What comes next in this sequence?

1, 3, 9

Column A

Column B

1, 4, 9, 16...
50
2, 8, 18, 32...
25
0, 3, 8, 15...
24

What is the seventh term in the sequence?

2, 5, 10, 17....

For each of these sequences, select if it is a geometric, a quadratic or neither.

 GeometricQuadraticNeither
1 4 9 16...
1 4 7 10...
3 12 27 64
40 20 10....

Which number could come next in this sequence?

 

1, 4....

What are we multiplying by each time to generate the sequence...

40, -20. 10. -5

 

Match the sequence with its type.

Column A

Column B

2,5,8,11
Geometric
1,5,25
Linear
1,4,9,16
Quadratic
  • Question 1

Complete the following sentence...

CORRECT ANSWER
EDDIE SAYS
This is one for the definition lovers. It\'s called a common ratio which means we have to multiply.
  • Question 2

Match these sequences with the correct common ratio

CORRECT ANSWER

Column A

Column B

1, 2, 4, 8
x 2
100, 10, 1
x 0.1
5, 15, 75
x 3
10, 10, 10, 10
x 1
EDDIE SAYS
All we need to do is ask what we're multiplying by. Remember that a division can also be written as a divide ÷10 is the same as x 1/10 (1/10 is the same as 0.1)
  • Question 3

Complete the following sentence

CORRECT ANSWER
EDDIE SAYS
This is one for the definition lovers. A second difference means we have a quadratic sequence
  • Question 4

Match the sequence to its next value.

CORRECT ANSWER

Column A

Column B

1, 4, 9, 16...
25
2, 8, 18, 32...
50
0, 3, 8, 15...
24
EDDIE SAYS
It's all about the second differences here, Remember to find the differences and then look for the pattern.
  • Question 5

What comes next in this sequence?

1, 3, 9

CORRECT ANSWER
EDDIE SAYS
A lovely little geometric sequence here. Each number is three times the previous one so we just multiply by 3.
  • Question 6

What is the seventh term in the sequence?

2, 5, 10, 17....

CORRECT ANSWER
50
EDDIE SAYS
With quadratic sequences, you can always look to see a little shortcut. Did you notice that each number here is one more than a square number? The seventh square number is 49, so our seventh number is 50
  • Question 7

For each of these sequences, select if it is a geometric, a quadratic or neither.

CORRECT ANSWER
 GeometricQuadraticNeither
1 4 9 16...
1 4 7 10...
3 12 27 64
40 20 10....
EDDIE SAYS
Remember to look at the differences It's geometric if it is multiplied (or divided) It's quadratic if the second differences are common
  • Question 8

Which number could come next in this sequence?

 

1, 4....

CORRECT ANSWER
16
7
9
EDDIE SAYS
There are a few that could be right here. If it is geometric then we multiply by 4 to get 16 If it is linear then we add 3 to get 7 We could also say it is quadratic and the next number would be 9 All off these would be fine.
  • Question 9

What are we multiplying by each time to generate the sequence...

40, -20. 10. -5

 

CORRECT ANSWER
-0.5
-1/2
EDDIE SAYS
For each term we are dividing by 2. This is the same as multiplying by 0.5 (or 1.2 if you prefer). The only way we can change from a positive to a negative to a positive etc is to multiply by -0.5
  • Question 10

Match the sequence with its type.

CORRECT ANSWER

Column A

Column B

2,5,8,11
Linear
1,5,25
Geometric
1,4,9,16
Quadratic
EDDIE SAYS
Remember the rules. Increase by the same amount - Linear Multiply by the same amount - Geometric \"nd difference is common - Quadratic
---- OR ----

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