Test

Question Type 11: Number Series

Eleven-Plus Preparation Specialists

One of three types of question that involves no letters whatsoever, this is purely a test of your reasoning ability – and indeed of your ability with numbers. Note that this question might be present in some papers (e.g. GL) in the Verbal Reasoning section, while it will be in the Maths section for other exam boards.

What does this question type involve?

This question simply requires you to be able to follow a sequence of numbers and predict what will come next. You will be provided with perhaps five different numbers, and then asked which number will come next. You could also be asked which number comes at a different point in the series (e.g. you could be asked the first number and be given five numbers after it).

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Example questions

Please find the next number in the series.

1,3,5,7,9,?

Answer: 11

This is a simple one. You simply have to add +2 each time to get to the nex number.

1,4,9,16,25,?

Answer: 36
Here, the next number is a square number each time. So the next one must be 6*6, which is 36.

4 Step Approach to Find the Next Number

We’ll avoid using algebra here, as the patterns are unlikely to be complex enough to require it – and this is a verbal reasoning paper, rather than maths. As such, let’s focus on the following steps:

Step 1: Linear addition
Step 2: Nonlinear addition
Step 3: Previous Numbers
Step 4: Squares

Our first step is linear addition. That means adding the same number each time. So, look at the pattern and see if it looks like a constant number is being added each time. If it is, then you should be able to quickly solve the question, as this is a linear sequence. An example would be adding +2 each time, for example.

After non-linear addition comes nonlinear addition. So, we need to see if the number changes by a predictable, yet different, number each time. So, the sequence could be +1,+2,+3, for example. Another common example is +2,+4,+6 etc.

If we still don’t see a clear pattern, then we should try a common pattern, which is the ‘previous numbers’ pattern. This sees two numbers being added together to give the next number, e.g.
1,2,3,5,8,13,21,34

If we still cannot see a pattern, check to see if squares are involved. This might be the simple pattern above, which is each of the smallest squares in order. 

Worked Example

Please find the next number in the series.

0, -3, -8, -15,?

So, we begin with linear addition. It should quickly be obvious that this is nonlinear, as the difference between the numbers is not constant throughout the series – as such, we need to move onto the next step in the process. Is there a simple nonlinear addition? Looking at it, we find that the difference between two numbers is 3, then 5, then 7. So, it seems we’ve solved the question, as simply take two more each time, beginning with -3. In fact, the thought behind this pattern was that it was 1 – each square number in turn. However, you can solve it much more simply just by thinking of it as a nonlinear sequence where you subtract two more each time.

This is an important point to remember, of course. The sequences may go down as well as up, and you may need to find lower numbers rather than just the number at the end – so be ready to subtract as well as add.

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Top Tricks

There are few tricks for this question – you just need to follow through a logical process and keep it simple. Always begin by looking for linear patterns, then focus on nonlinear patterns next. Don’t be thrown if the pattern seems to be entirely random – it isn’t, and if you work through the steps above you should find it.

Common Pitfalls

A common error is being confused by series that ask you to find a lower number. If, for example, you’re asked to find the first number in a series, then just see the series in reverse, and work from back to front.

Summary

In summary, this is a simple mathematical question that shouldn’t ask for too complex a sequence.
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