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 DiagramsBelow are the first four triangle numbers, represented in diagrams:
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:
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. 
The next three differences must therefore be 8, 9 and 10. This means that:
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. To mark a point on the grid, click it. To remove the mark, click the same place again.
Produced by A.J. Reynolds January 2001
Source: https://www.cimt.org.uk/projects/mepres/book7/bk7i13/bk7_13i2.htm