Unit 13 Section 2 : Extending Number Sequences

The idea of extending sequences was first covered in Unit 7.
This section takes these ideas and extends them to include some other types of sequences.

Extending Triangle Numbers

Using Diagrams
Below are the first four triangle numbers, represented in diagrams:
They are called triangle numbers because they correspond to the number of dots in a triangle.

If we want to find out what the next three triangle numbers are, we can draw more diagrams.
Each new diagram adds one more row to the triangle, with one more dot in it:
We can now see that the first seven triangle numbers are:

1, 3, 6, 10, 15, 21, 28,

Using Differences
It is possible to extend this sequence without using diagrams.
We need to look at the difference between the terms in the sequence.
You should notice that the difference between each term increases by 1 as you move along the sequence.

The next three differences must therefore be 8, 9 and 10. This means that:
  • the 8th term will be 28 + 8 = 36
  • the 9th term will be 36 + 9 = 45
  • the 10th term will be 45 + 10 = 55
So the first ten triangle numbers are:

1, 3, 6, 10, 15, 21, 28, 36, 45, 55,

Example Question

We want to write down the first ten terms of the sequence 3, 4, 7, 12, 19, 28, 39, ...

(a) Work out the differences between each term, then click Click on this button below to see the correct answer to see whether you are correct.

(b) Now work out what the 8th, 9th and 10th terms are, by continuing the pattern.

Other Sequences

These are some well-known sequences you should be aware of:

 

Exercises

Work out the answers to the questions below and fill in the boxes. Click on the Click this button to see if you are correct button to find out whether you have answered correctly. If you are right then will appear and you should move on to the next question. If appears then your answer is wrong. Click on to clear your original answer and have another go. If you can't work out the right answer then click on Click on this button to see the correct answer to see the answer.

Question 1
Work out the next four terms of each of the sequences below. Separate the terms with commas, like this:
(a) 4, 7, 10, 13, 16, 19, ...
(b) 5, 11, 17, 23, 29, 35, ...
(c) 6, 8, 11, 15, 20, 26, ...
(d) 8, 10, 14, 20, 28, 38, ...
(e) 24, 23, 21, 18, 14, 9, ...
(f) 2, 12, 21, 29, 36, 42, ...
(g) 1, 1, 2, 4, 7, 11, ...

For some of the next few questions, you need to illustrate patterns on a grid.
To mark a point on the grid, click it. To remove the mark, click the same place again.
Question 2
The diagram below shows the first 4 square numbers.
(a) Draw the next two square numbers on the grid to the right.
(i) What is the 5th square number?
(ii) What is the 6th square number?
(b) Without drawing further patterns, try and work out:
(i) the 10th square number
(ii) the 20th square number
(c) Look at the differences between each of the terms.
What will the difference between the 6th and 7th terms be?

Question 3
Write down the next three terms in each of these sequences.
(a) 0, 3, 8, 15, 24, ...
(b) 2, 5, 10, 17, 26, ...
(c) 11, 14, 19, 26, 35, ...
(d) 6, 9, 14, 21, 30, ...
(e) The difference patterns in these sequences are all the same as one of the special sequences. Which one?

Question 4
In each of the sequences below, use the grid provided to draw the next two patterns.
Fill in the box with the first 8 terms of the sequence represented by the number of dots in each pattern.
(a)
(b)
(c)

Question 5
In each of the sequences below, use the grid provided to draw the next three patterns.
Fill in the box with the first 8 terms of the sequence represented by the number of dots in each pattern.
(a)

(b)

(c)

Question 6
Work out the 10th number in each of the sequences illustrated by the diagrams below.
You may find it helpful to draw the 10th diagram on squared paper.
(a)
The tenth number is
(b)
The tenth number is
(c)
The tenth number is

Question 7
The Fibonacci sequence begins 1, 1, 2, 3, 5, 8, ... and each term is found by adding the two previous terms together.

(a) What is the 10th term in the sequence?

(b) What is the 20th term in the sequence?

Question 8
Write down the next 4 terms in each of these sequences:
(a) 2, 2, 4, 6, 10, ...
(b) 1, 3, 4, 7, 11, ...
(c) 2, 5, 7, 12, 19, ...
(d) 1, 9, 10, 19, 29, ...

You have now completed Unit 13 Section 2
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Produced by A.J.Reynolds January 2001