Draw Three Diagram to Continue This Dot Sequence

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:

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 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

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

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 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 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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Source: https://www.cimt.org.uk/projects/mepres/book7/bk7i13/bk7_13i2.htm