 # Recognise Quadratic and Geometric Sequences

In this worksheet, students will recognise all three types of sequences (linear, geometric and quadratic), find the rules which govern them, and apply these to find missing terms. Key stage:  KS 4

GCSE Subjects:   Maths

GCSE Boards:   AQA, Eduqas, OCR, Pearson Edexcel

Curriculum topic:   Algebra

Curriculum subtopic:   Sequences

Difficulty level:   ### QUESTION 1 of 10

There are a number of different types of sequences, some of which you may already be familiar with or have completed activities to learn more about.

The most common sequences are:

Linear: A sequence that increases or decreases by the same amount each time.

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

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

Let's see these sequences in practice now.

e.g. Describe the rule for, and give the next two terms of, the sequence 3, 6, 12, 24, ...

We should notice that the sequence here is doubling each time

This means the rule (or the common ratio) is × 2.

If we then continue the sequence using this rule, the next two terms will be:

5th: 24 × 2 = 48

6th: 48 × 2 = 96

e.g. 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 or down by the same amount so it can't be linear, but it also isn't multiplying each time, so it can't be a geometric sequence either.

This means it must be a quadratic sequence.

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 that the difference between the fifth and sixth terms is 11.

If we add this to the fifth term, we will find our next term:

27 + 11 = 38

In this activity, we will need to recognise all three types of sequences, find the rules which govern them, and apply these to find missing terms.

You should not use a calculator in this activity, but instead practise your mental arithmetic.

Complete the sentence below to define a geometric sequence

Match each sequence below with its correct common ratio.

## Column B

1, 2, 4, ...
× 0.1
100, 10, 1, ...
× 2
5, 15, 45, ...
× 3
10, 10, 10, ...
× 1

Complete the sentence below to define a quadratic sequence

## Column B

1, 2, 4, ...
× 0.1
100, 10, 1, ...
× 2
5, 15, 45, ...
× 3
10, 10, 10, ...
× 1

Match each sequence below to its correct next term

## Column B

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

What are the next two terms in this sequence?

1, 3, 9, __, __

## Column B

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

What is the seventh term in this sequence?

2, 5, 10, 17, ...

For each of the sequences below, select its correct type from the options below.

Choose 'Neither' if you do not think it is quadratic or geometric.

 Geometric Quadratic Neither 1 4 9 16 ... 1 4 7 10 ... 3 12 27 64 ... 40 20 10 5 ...

Which number could come next in this sequence?

1, 4, ...

16

15

10

7

8

9

What are we multiplying by each time to generate this sequence?

40, -20, 10, -5, ...

Match each of the sequences below to its correct type from the options.

## Column B

2, 5, 8, 11, ...
Geometric
1, 5, 25, 125, ...
Linear
1, 4, 9, 16, ...
• Question 1

Complete the sentence below to define a geometric sequence

EDDIE SAYS
Let's lock these important definitions down before we move on to the rest of this activity. A geometric sequence has a common ratio, which means it increases or decreases by the same factor each time. This means that we have to multiply to continue the sequence. An example of a geometric sequence would be: 3, 6, 12, 24, ... Which has a common ratio of × 2.
• Question 2

Match each sequence below with its correct common ratio.

## Column B

1, 2, 4, ...
× 2
100, 10, 1, ...
× 0.1
5, 15, 45, ...
× 3
10, 10, 10, ...
× 1
EDDIE SAYS
All we need to do here is to ask ourselves what we're multiplying by in each case. Our × 2 and × 3 sequences are relatively simple, but did you spot that the sequence which remained the same, had a common ration of × 1? Remember that when a sequence is reducing by a common multiple or dividing, we can still write this in the format '× ...' using a decimal: ÷ 10 is the same as × 1/10. 1/10 is the same as 0.1. Did you match those sequences and ratios correctly?
• Question 3

Complete the sentence below to define a quadratic sequence

EDDIE SAYS
Let's lock these important definitions down before we move on to the rest of this activity. A quadratic sequence has a common second difference, which means that the difference between its differences are the same. An example of a quadratic sequence would be: 4, 6, 10, 16, 24, ... Which has a common second difference of + 2.
• Question 4

Match each sequence below to its correct next term

## 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, as we are dealing with quadratic sequences. We need to find the differences between each term and then look for the pattern. Let's look at the first sequence together: 1, 4, 9, 16, ... The differences here are: + 3, + 5, + 7, etc. Can you see that these differences are increasing by + 2 each time? So we need to add 2 more to this pattern to find the fifth term. 5th: 16 + (7 + 2) = 25 Can you follow this example to match the other two pairs independently?
• Question 5

What are the next two terms in this sequence?

1, 3, 9, __, __

EDDIE SAYS
We have a lovely little geometric sequence here! Each term is three times the previous one. So to find the next terms, we just multiply the previous one by 3. 4th: 9 × 3 = 27 5th: 27 × 3 = 81
• Question 6

What is the seventh term in this sequence?

2, 5, 10, 17, ...

50
EDDIE SAYS
This is a quadratic sequence. The differences between these terms are: + 3, + 5, + 7, ... Can you see that the differences are increasing by 2 each time? So to find the next few terms, we need to apply this rule: 5th: 17 + (7 + 2) = 26 6th: 26 + (9 + 2) = 37 5th: 37 + (11 + 2) = 50 With quadratic sequences, you can always look to see a little shortcut too. Did you notice that each number here is one more than a square number? The seventh square number is 49, so our seventh number in this sequence must be 50.
• Question 7

For each of the sequences below, select its correct type from the options below.

Choose 'Neither' if you do not think it is quadratic or geometric.

 Geometric Quadratic Neither 1 4 9 16 ... 1 4 7 10 ... 3 12 27 64 ... 40 20 10 5 ...
EDDIE SAYS
Remember to look at the differences between the terms here. It's geometric if it is multiplied (or divided) by a common ratio. It's quadratic if the second differences have a common second difference. Only the second option is neither quadratic or geometric. This sequence is increasing by + 3 each time, which make it a linear sequence. Are you feeling more confident to apply these definitions now?
• Question 8

Which number could come next in this sequence?

1, 4, ...

16
7
9
EDDIE SAYS
There are a few answers that could be right here. If it is geometric sequence, then we multiply by 4 to find the next term of 16. If it is a linear sequence, then we could add 3 to find the next term of 7. We could also say it is a quadratic sequence, in which case, our next term would be 9. Therefore, all these three answers are correct. We would need to see how the sequence develops in order to know for sure that only one of these options is, in fact, correct. For this reason, you are unlikely to see such short sequences in exam questions, as there are often many possible answers.
• Question 9

What are we multiplying by each time to generate this sequence?

40, -20, 10, -5, ...

-0.5
- 0.5
-1/2
- 1/2
EDDIE SAYS
To find the next terms here, we are dividing by 2. This is the same as multiplying by 0.5 or 1/2. The only way we can create a sequence which alternates between positive and negative numbers, is to multiply by a negative ratio. So the correct answer here is -0.5 or -1/2. Does that make sense?
• Question 10

Match each of the sequences below to its correct type from the options.

## Column B

2, 5, 8, 11, ...
Linear
1, 5, 25, 125, ...
Geometric
1, 4, 9, 16, ...
EDDIE SAYS
Remember our sequence types to apply here: Linear sequences increase/decrease by the same amount. Geometric sequences multiply by the same amount. Quadratic sequences apply an increase/decrease in the second differences of the same amount. Did you match these correctly here? Congratulations on completing this activity! If you are feeling confident, why not try a Level 3 sequences activity to stretch your abilities further?
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