Number series questions in numerical reasoning assessments consist of a group of numbers that share a logical rule based on elementary arithmetic.

Numbers in a series are described as ‘terms’, and they are presented in a sequence with gaps – the candidate must find the missing term(s) to complete the question. As these are part of numerical reasoning tests, the answers are usually presented in a multiple-choice format.

In the sequence, there are typically between four to seven visible numbers, and to find the missing elements the candidate needs to find the specific logical rule or pattern that links the series, and apply that to the missing numbers.

Number series questions are based on rather simple mathematical rules. They are difficult because the pattern may be less obvious, rather than it being based on complicated arithmetic.

## Different forms of number series questions

There are several types of number series questions, and the types are based on what logic connects the numbers in a sequence.

### Arithmetic sequence

The relationship between terms in an arithmetic number series is based on adding, subtracting, or a combination of both operations. The most basic version of this is 1, 2, 3, 4, 5 __ where the next number would be six because the rule is to add one to the last digit.

### Geometric sequence

In a similar way to arithmetic sequences, geometric sequences contain numbers that share a rule based on multiplication or division.

### Mixed sequence

A mixed sequence describes the relationship between terms as being based on more than one rule. This can make finding the right answer more complicated, as the functions might not be from the same family – such as a combination of arithmetic and geometric sequences.

### Alternating sequence

An alternating sequence is a single number series made of two independent sub-sequences. This can make finding the rule or pattern complicated, since the numbers are not necessarily related to the terms on either side in the series itself.

### Fibonacci sequence

The Fibonacci sequence makes the terms related to one another by adding the two previous numbers together. The usual Fibonacci sequence is 1, 1, 2, 3, 5, 8, 13, 21 and so on, but there are no guarantees that the sequence will start at one or even be an ascending series.

There are several other types of sequence too, including prime numbers, perfect square and perfect cube series.

## Five example number series questions

### Example arithmetic sequence:

1, 5, 9, 13, 17, _

What is the next number in this series?

- 15
- 23
- 21
- 20

The rule for this pattern is to add 4 to the previous number, so in this case, the answer would be c) 21.

### Example geometric sequence:

Find the missing number in the series.

120, 60, 30, __, 7.5, 3.25

- 20
- 5
- 18
- 15

This sequence is solved in the same way as above, even though the missing number is in the middle. The relationship between each number is the division by two of the previous number – and it’s important to understand that the terms in a series don’t need to be integers. In this question, the missing term is d) 15.

### Example mixed sequence:

What is the next number in this sequence?

2, 5, 11, 23, 47, __

- 95
- 101
- 94
- 97

This pattern combines geometric and arithmetic sequences, and the rule is that each number is the previous number multiplied by two, and then add one.

The difficulty here is establishing the right combination of mathematical functions that are needed. In this example, the next term in the series would be 95, so the answer is a).

### Example alternating sequence:

Find the next two terms in the series.

7, 18, 14, 16, 28, 14, __, __

- 57, 12
- 56, 12
- 58, 10
- 56, 10

The alternating sequence needs the candidate to make two separate calculations, as each number is not related to the ones immediately before or following.

The first number and every other number are related by multiplying 2 to the previous, and the second number and every other is based on a simple subtraction of one. This means that the next two numbers in the series are b) 56, 12.

### Example Fibonacci sequence:

Find the missing term.

4, 4, 8, 12, __, 32, 52

- 22
- 24
- 20
- 18

Recognising a Fibonacci sequence makes it simple to find the rule; in this case, it is easy to ascertain that the terms are the sum of the previous two numbers.

Again, even though the missing number is in the middle of the series, the methodology for finding the answer remains the same. The previous two terms for the missing number are 8 and 12, which makes the answer c) 20.

## Tips for working out number series questions

### 1) Practice

Working on number sequences becomes easier when you have practiced; recognising common types of sequence is easier if you are familiar with what they tend to look at. You can find number sequence practice in our numerical reasoning tests guide.

### 2) Remember basic maths

The mathematical skills that are needed to make the connection between terms are not difficult. Brushing up on basic arithmetic knowledge is all that is needed to give you confidence for completing number series questions.

### 3) There may be more than one way to solve

There could be more than one logical relationship between terms in the pattern. This can make it easier to solve the series and gives you another way to find the answer.

### 4) Check the difference between terms

If you are struggling to find the relationship between terms in a sequence, looking at the difference between each number might give you the clue you need.

### 5) You can estimate

Time is of the essence when you are in a numerical reasoning assessment, so if you don’t immediately know the answer it’s fine to make an educated guess. In most cases you will not have a calculator, so if the terms are larger estimating might be faster.