Recognise Other Types of Sequences

In this worksheet, students will recognise and define the types of sequences shown, as well as finding unknown terms by continuing sequences in the same pattern.

Key stage:  KS 4

GCSE Subjects:   Maths

GCSE Boards:   AQA, Eduqas, Pearson Edexcel, OCR

Curriculum topic:   Algebra

Curriculum subtopic:   Sequences

Difficulty level:

QUESTION 1 of 10

You may already be familiar with recognising linear sequences (such as 4, 7, 10, 13, ...) which increase or decrease by the same amount each time.

However, if we look at the definition of a sequence, (a pattern of numbers that follow the same rule), there are many other different types of sequences which follow different rules which it is helpful to be able to recognise too.

Fibonacci sequences: These sequences are generated by adding the previous two numbers in the sequence.

e.g. 1, 1, 2, 3, 5, 8, 13, 21, ...

Square numbers: These sequences are created by squaring the position of the number. So the first number is 1 × 1, the second is 2 × 2, etc.

e.g. 1, 4, 9, 16, 25, 36, 49, ...

Powers sequences: These sequences are created using the power of a number. So the powers of 3, would be 31, 32, 33, 34, 35, ...:

e.g. 3, 9, 27, 81, 243, ...

These sequences can be spotted quite easily, as each number is a multiple of the previous one (i.e. 9 is 3 × 3; 27 is 9 × 3; 81 is 27 × 3; etc.)

Triangular numbers: These are a bit harder to spot, as each time the difference increases by 1.

e.g. 1, 3, 6, 10, 15, ...

1 to 3 = difference of 2; 3 to 6 = difference of 3; 6 to 10 = difference of 4; 10 to 15 = difference of 5, etc.

In this activity, we will recognise and define the types of sequences shown, as well as finding unknown terms by continuing sequences in the same pattern.

Complete the sentence below to define Fibonacci sequences.

Select the correct type from the options below for this sequence.

3, 4, 7, 11, 18, ...

Fibonacci

Power

Square

Triangular

Select the correct type from the options below for this sequence.

2, 8, 19, 32, ...

Fibonacci

Power

Square

Triangular

Select the correct type from the options below for this sequence.

2, 4, 8, 16, 32, ...

Fibonacci

Power

Square

Triangular

Which two numbers comes next in this sequence?

1, 1, 2, 3, 5, 8, 13, ...

Fibonacci

Power

Square

Triangular

Which two numbers comes next in this sequence?

243, 81, 27, ...

Fibonacci

Power

Square

Triangular

Match each sequence below with its correct type.

Match each sequence below with its correct next term.

Column B

2, 3, 5, 8, 12, ...
13
1, 1, 2, 3, 5, 8, ...
36
1, 4, 9, 16, 25, ...
17

Complete the statement below to describe how to recognise a powers sequence.

Column B

2, 3, 5, 8, 12, ...
13
1, 1, 2, 3, 5, 8, ...
36
1, 4, 9, 16, 25, ...
17

Complete the sentence below to define the given sequence.

Column B

2, 3, 5, 8, 12, ...
13
1, 1, 2, 3, 5, 8, ...
36
1, 4, 9, 16, 25, ...
17
• Question 1

Complete the sentence below to define Fibonacci sequences.

EDDIE SAYS
Did you remember the definition of a Fibonacci sequence from the Intro? These sequences are generated by adding the previous two numbers in the sequence. Take a moment to review the other definitions of sequences in the Introduction before you move on to tackle the rest of this activity.
• Question 2

Select the correct type from the options below for this sequence.

3, 4, 7, 11, 18, ...

Fibonacci
EDDIE SAYS
The only option this fits this sequence is Fibonacci. Let's look at a number then add the two before it; every number is a total of the two numbers preceding it. Did you spot our definition from Q1 in action here?
• Question 3

Select the correct type from the options below for this sequence.

2, 8, 19, 32, ...

Square
EDDIE SAYS
This sequence has to be using a square rule. If we think about the square numbers (1, 4, 9, 16, etc.), can you see the link between these and our sequence? That's right, our sequence shows doubles of our square numbers.
• Question 4

Select the correct type from the options below for this sequence.

2, 4, 8, 16, 32, ...

Power
EDDIE SAYS
Each number in this sequence is double the one before; this means it is a powers sequence. Each term we are seeing here is a power of 2: 21, 22, 23, 24, 25, etc. These sequences can be spotted quite easily, as each number is a multiple of the previous one.
• Question 5

Which two numbers comes next in this sequence?

1, 1, 2, 3, 5, 8, 13, ...

EDDIE SAYS
Our first step here is to identify the type of sequence shown. This is a Fibonacci sequence, as each number is calculated by adding the two numbers together which precede it. So to get the next two numbers, we add the last two numbers together: 8 + 13 = 21 And again: 13 + 21 = 34 Did you type those values in correctly?
• Question 6

Which two numbers comes next in this sequence?

243, 81, 27, ...

EDDIE SAYS
Our first step is to identify the type of sequence shown here. As we are dividing by 3 each time, this has to be a powers sequence. Each term we are seeing here is a power of 3: 35, 34, 33, 32, 31, etc. We just need to keep dividing each of the previous terms by 3 to get the next two answers: 27 ÷ 3 = 9 9 ÷ 3 = 3
• Question 7

Match each sequence below with its correct type.

EDDIE SAYS
Try and use a process of elimination here. Powers are usually the easiest sequences to spot as they involve numbers which we are often familiar with. Can you spot the sequence which uses powers of 7? Squares are also relatively easy to pick out, as they must contain our known square numbers or multiples of these. Can you see which sequence involves the square numbers multiplied by 2? Fibonacci sequences can be spotted by adding together two numbers and seeing if we reach the next. Triangular sequences will be the sequence that doesn't fit with any of the other types.
• Question 8

Match each sequence below with its correct next term.

Column B

2, 3, 5, 8, 12, ...
17
1, 1, 2, 3, 5, 8, ...
13
1, 4, 9, 16, 25, ...
36
EDDIE SAYS
Let's define each sequence and then use this definition to help us find the next term. 2, 3, 5, 8, 12, ... is a triangular sequence, so we can find our next term as each time the difference increases by 1: 2 to 3 = difference of 1; 3 to 5 = difference of 2; 5 to 8 = difference of 3; 8 to 12 = difference of 4; 12 to ? = difference of 5: So 12 + 5 = 17 1, 1, 2, 3, 5, 8, ... is a Fibonacci sequence, so we can find our next term by adding the two previous terms: 5 + 8 = 13 1, 4, 9, 16, 25, ... is a square sequence, so we can find our next term by finding the next square number based on position: 6 × 6 = 36
• Question 9

Complete the statement below to describe how to recognise a powers sequence.

EDDIE SAYS
Power sequences can be spotted because we need to multiply the same number by itself to reach the next term in the sequence. This means each term is a multiple of the previous one.
• Question 10

Complete the sentence below to define the given sequence.

EDDIE SAYS
This is one of those tricky Fibonacci sequences. Remember that the way we test this is to add together two terms and see if we reach the next term. If the sequence follow this pattern, then it's a Fibonacci type. Amazing work on this activity! Hopefully you can spot those different types of sequences easily now.
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