# Types of Number series

## What is number series?

It is a sequence of numbers that follow a particular pattern, i.e. there is an underlying logic/rule that all elements of that series have to follow.

For example, 1, 2, 3, 4, 5 ….. (each subsequent number in this series is 1 more than the previous number)

You will be much more comfortable in solving such questions, if you acclimatize yourself with numbers. You should know the following to be good in this chapter:

- Squares till 32
- Cubes till 12
- Few concepts of Number System – even/odd, prime numbers etc.
- Few other miscellaneous concepts like Roman Numbers, Fibonacci series etc.

There are two broad categories of questions that are asked in number series:

- Find the missing number in the series
- Find the wrong number in the series

Let’s study them, one by one.

## Finding Missing Number in the Number series

Different logics are used to formulate such questions. We should be familiar will most of them, so that we may be able to recognize them in the exam time.

### Type 1: Same Number is added/subtracted

In this type of series, the difference between two consecutive numbers is the same.

For example, 2, 4, 6, 8, 10, ?

Each subsequent number in this series is 2 more than the previous number. So, we are adding 2 to get the next number in the series.

Series which looks addition type in one direction will look subtraction type in reverse direction and vice-versa.

For example, in the previous series: 2, 4, 6, 8, 10, ?

If we move from right to left, then we can say that each subsequent number in this series is 2 less than the previous number. So, we are subtracting 2 to get the next number in the series.

##### Q. Find the missing number in the following series:

0.5, 2, 3.5, 5, 6.5, 8, ?

(a) 9.5 (b) 10.5 (c) 9.0 (d) 11.0

Explanation:

Pattern:

0.5 + 1.5 = 2

2 + 1.5 = 3.5

3.5 + 1.5 = 5

5 + 1.5 = 6.5

6.5 + 1.5 = 8

8 + 1.5 = **9.5**

Next element = Previous element + 1.5

Answer: (a)

##### Q. Find the missing number in the following series:

110, 107, 104, ?, 98, 95

(a) 102 (b) 105 (c) 100 (d) 101

Explanation:

Rule:

Next element = Previous element - 3

So, missing number = **101**

Answer: (d)

Sometimes, you may be given numbers in some other form, e.g. Roman numbers. So, you must have familiarity with them.

##### Q. Find the missing number in the following series:

II, V, VIII, XI, XIV, ?, XX

(a) XVI (b) XVII (c) XV (d) XIX

Explanation:

Pattern:

II + 3 = V

V + 3 = VIII

VIII + 3 = XI

XI + 3 = XIV

XIV + 3 = **XVII**

XVII + 3 = XX

(i.e. Next element = Previous element + 3)

Answer: (b)

Sometimes, couple of numbers may be added/subtracted in alternate fashion.

##### Q. Find the missing number in the following series:

30, 28, 23, 21, ?

(a) 17 (b) 16 (c) 19 (d) 15

Explanation:

Pattern:

30 - 2 = 28

28 - 5 = 23

23 - 2 = 21

21 - 5 = **16**

Answer: (b)

### Type 2: Number added/subtracted is constantly increased/decreased

In this type of series, number added or subtracted to each term is in increasing or decreasing order.

##### Q. Find the missing number in the following series:

3, 5, 8, 12, 17, ?

(a) 25 (b) 23 (c) 19 (d) 21

Explanation:

In the given ascending series, the difference between two consecutive numbers is in increasing order i.e., 2, 3, 4, 5 and 6, respectively.

Pattern:

3 + 2 = 5

5 + 3 = 8

8 + 4 = 12

12 + 5 = 17

17 + 6 = **23**

Answer: (b)

##### Q. Find the missing number in the following series:

30, 24, 19, 15, ?

(a) 13 (b) 11 (c) 12 (d) 10

Explanation:

In the given ascending series, the difference between two consecutive numbers is in decreasing order i.e., 6, 5, 4, and 3, respectively.

Pattern:

30 - 6 = 24

24 - 5 = 19

19 - 4 = 15

15 - 3 = **12**

Answer: (c)

### Type 3: Same Number is multiplied/divided

In this type of series, the ratio between two consecutive numbers is the same. That is, we divide or multiply the previous element by the same number to get the next element in the series.

Series which looks multiplication type in one direction, will look division type in another direction and vice-versa.

##### Q. Find the missing number in the following series:

4, 12, 36, 108, ?

(a) 180 (b) 324 (c) 264 (d) 216

Explanation:

In the given ascending series, the ratio between two consecutive numbers is 3. So, it’s basically a simple multiplication series, if seen from left to right.

Pattern:

4 × 3 = 12

12 × 3 = 36

36 × 3 = 108

108 × 3 = **324**

Answer: (c)

##### Q. Find the missing number in the following series:

168, 84, 42, 21, ?

(a) 11 (b) 9.5 (c) 10 (d) 10.5

Explanation:

In the given descending series, the ratio between two consecutive numbers is 2. So, it’s basically a simple division series, if seen from left to right.

Pattern:

168 ÷ 2 = 84

84 ÷ 2 = 42

42 ÷ 2 = 21

21 ÷ 2 = **10.5**

Answer: (d)

Sometimes, multiplication/division may be done by a couple of numbers, in alternate fashion.

##### Q. Find the missing number in the following series:

24, 6, 18, 9, 36, 9, 24, ?

(a) 12 (b) 8 (c) 6 (d) 30

(SSC Question)

Explanation:

Pattern:

24 ÷ 4 = 6

18 ÷ 2 = 9

36 ÷ 4 = 9

24 ÷ 2 = **12**

Answer: (a)

### Type 4: Increasing order Multiplication / Division

In this type of series, elements are multiplied or divided with numbers in increasing/decreasing order to obtain the next element.

##### Q. Find the missing number in the following series:

3, 3, 4.5, 9, 22.5, ?

(a) 24 (b) 27.5 (c) 67.5 (d) 37.5

Explanation:

Pattern:

3 × 1 = 3

3 × 1.5 = 4.5

4.5 × 2 = 9

9 × 2.5 = 22.5

22.5 × 3 = **67.5**

Answer: (c)

##### Q. Find the missing number in the following series:

720, ?, 36, 12, 6, 6

(a) 108 (b) 144 (c) 360 (d) 72

(SSC Question)

Explanation:

Pattern:

720 ÷ 5 = **144**

144 ÷ 4 = 36

36 ÷ 3 = 12

12 ÷ 2 = 6

6 ÷ 1 = 6

It’s easier to deal with smaller numbers. So, we may start from the right in the given series, to decipher the pattern. Also, as the missing number is right at the beginning, it makes sense to look at this series from right to left.

If seen from right to left, then this series will be an ascending multiplication series.

Answer: (c)

### Type 5: Mixed Series

In such series we see a mix of operators being used. That is, instead of just using a single addition, subtraction, multiplication or division operator throughout the series, we use a mix of them. So, obviously the difficulty level goes up.

Let’s have a look at some examples. It will make this more clear to you.

#### Type 5a: Same multiplication, Same addition

##### Q. Find the missing number in the following series:

5, 11, 23, 47, 95, ?

(a) 185 (b) 191 (c) 192 (d) 188

Explanation:

Next number = Previous number × 2 + 1

Pattern:

(5 × 2) + 1 = 11

(11 × 2) + 1 = 23

(23 × 2) + 1 = 47

(47 × 2) + 1 = 95

(95 × 2) + 1 = **191**

Another way to look at it:

The differences between the elements in the series are:

6, 12, 24, 48, 96

(i.e. difference doubles each time)

Answer: (a)

#### Type 5b: Same multiplication, Increasing addition

##### Q. Find the missing number in the following series:

5, 11, 24, 51, 106, ?

(a) 178 (b) 217 (c) 185 (d) 191

Explanation:

Pattern:

(5 × 2) + 1 = 11

(11 × 2) + 2 = 24

(24 × 2) + 3 = 51

(51 × 2) + 4 = 106

(106 × 2) + 5 = **217**

Answer: (b)

#### Type 5c: Increasing multiplication, Same addition

##### Q. Find the missing number in the following series:

3, 6, 15, 48, ?

(a) 110 (b) 99 (c) 195 (d) 191

Explanation:

Pattern:

(3 × 1) + 3 = 6

(6 × 2) + 3 = 15

(15 × 3) + 3 = 48

(48 × 4) + 3 = **195**

Answer: (c)

#### Type 5d: Increasing multiplication, Increasing addition

##### Q. Find the missing number in the following series:

4, 5, 12, 39, 160, ?

(a) 325 (b) 805 (c) 555 (d) 705

Explanation:

Pattern:

(4 × 1) + 1 = 5

(5 × 2) + 2 = 12

(12 × 3) + 3 = 39

(39 × 4) + 4 = 160

(160 × 5) + 5 = **805**

Answer: (b)

#### Type 5e: Same multiplication, Same subtraction

##### Q. Find the missing number in the following series:

4, 5, 7, 11, 19, 35, ?

(a) 55 (b) 62 (c) 67 (d) 71

Explanation:

Next element = Previous element × 2 – 3

Pattern:

(4 × 2) – 3 = 5

(5 × 2) – 3 = 7

(7 × 2) – 3 = 11

(11 × 2) – 3 = 19

(19 × 2) – 3 = 35

(35 × 2) – 3 = **67**

Another way to look at it:

The differences between the elements in the series are:

1, 2, 4, 8, 16, 32

(i.e. difference doubles each time)

Answer: (c)

#### Type 5f: Same multiplication, Increasing subtraction

##### Q. Find the missing number in the following series:

4, 7, 12, 21, 38, ?

(a) 71 (b) 67 (c) 78 (d) 81

Explanation:

Pattern:

(4 × 2) - 1 = 7

(7 × 2) - 2 = 12

(12 × 2) - 3 = 21

(21 × 2) - 4 = 38

(38 × 2) - 5 = **71**

Answer: (a)

#### Type 5g: Increasing multiplication, Same subtraction

##### Q. Find the missing number in the following series:

4, 6, 16, 62, ?

(a) 128 (b) 108 (c) 288 (d) 308

Explanation:

Pattern:

(4 × 2) - 2 = 6

(6 × 3) - 2 = 16

(16 × 4) - 2 = 62

(62 × 5) - 2 = **308**

Answer: (d)

#### Type 5h: Increasing multiplication, Increasing subtraction

##### Q. Find the missing number in the following series:

3, 5, 13, 49, ?

(a) 98 (b) 241 (c) 102 (d) 159

Explanation:

Pattern:

(3 × 2) - 1 = 5

(5 × 3) - 2 = 13

(13 × 4) - 3 = 49

(49 × 5) - 4 = **241**

Answer: (b)

#### Type 5i: Alternate usage of operators

Sometimes, we see alternate usage of addition and subtraction.

##### Q. Find the missing number in the following series:

5, 11, 21, 43, 85, ?

(a) 169 (b) 167 (c) 171 (d) 165

(SSC CGL Question)

Explanation:

1 is added and subtracted alternatively.

Pattern:

(5 × 2) + 1 = 11

(11 × 2) – 1 = 21

(21 × 2) + 1 = 43

(43 × 2) – 1 = 85

(85 × 2) + 1 = **171**

Answer: (c)

Similarly, sometimes we see alternate usage of multiplication and division.

##### Q. Find the missing number in the following series:

24, 72, 36, 108, 54, ?

(a) 98 (b) 27 (c) 108 (d) 162

Explanation:

Pattern:

24 × 3 = 72

72 ÷ 2 = 36

36 × 3 = 108

108 ÷ 2 = 54

54 × 3 = **162**

Answer: (d)

### Type 6: Square / Cube / Root Series

In this type of series, each element is the square, cube or root of a number in a certain sequence.

##### Q. Find the missing number in the following series:

4, 9, 16, 25, 36, ?

(a) 48 (b) 49 (c) 42 (d) 64

Explanation:

Pattern:

$2^2$ = 4

$3^2$ = 9

$4^2$ = 16

$5^2$ = 25

$6^2$ = 36

$7^2$ = **49**

Another way to look at it:

The differences between the elements in the series are:

5, 7, 9, 11, 13

(i.e. differences are in Arithmetic Progression)

Answer: (b)

##### Q. Find the missing number in the following series:

8, 27, 64, 125, ?

(a) 216 (b) 224 (c) 169 (d) 286

Explanation:

Pattern:

$2^3$ = 8

$3^3$ = 27

$4^3$ = 64

$5^3$ = 125

$6^2$ = **216**

Answer: (a)

##### Q. Find the missing number in the following series:

256, 16, 4, ?

(a) 6 (b) 8 (c) 2 (d) 1

(SSC LDC Question)

Explanation:

Pattern:

$\sqrt{256}$ = 16

$\sqrt{16}$ = 4

$\sqrt{4}$ = **2**

Answer: (c)

### Type 7: Square / Cube / Root Addition Series

In this type of series, square/cube/root of numbers are added to previous element in a particular manner to obtain the next element.

##### Q. Find the missing number in the following series:

2, 3, 7, 16, 32, ?

(a) 53 (b) 39 (c) 49 (d) 57

Explanation:

Pattern:

2 + $1^2$ = 3

3 + $2^2$ = 7

7 + $3^2$ = 16

16 + $4^2$ = 32

32 + $5^2$ = **57**

Answer: (d)

### Type 8: Prime Number Series

In this type of series, each element is a prime number in certain sequence.

##### Q. Find the missing number in the following series:

2, 3, 5, 7, 11, ?

(a) 15 (b) 19 (c) 13 (d) 21

Explanation:

The given number series is made up of consecutive prime numbers. The next prime number after 11 is 13.

Answer: (c)

### Type 9: Digital Manipulation based Number Series

In this type of series, the digits of each number are operated in a certain way to obtain the next element of the series.

##### Q. Find the missing number in the following series:

88, 64, 24, ?

(a) 12 (b) 8 (c) 16 (d) 15

Explanation:

Pattern:

8 × 8 = 64

6 × 4 = 24

2 × 4 = **8**

Answer: (b)

##### Q. Find the missing number in the following series:

121, 222, 424, ?

(a) 848 (b) 888 (c) 828 (d) 818

(SSC CGL Question)

Explanation:

In the given series, the first and the third digits get doubled each time.

Pattern:

121 → 222

222 → 424

424 → **828**

Answer: (c)

## Finding Wrong Number in the Number series

The candidate is required to identify the pattern involved in the formation of given series and then find out that number which does not follow the specific pattern of the series.

##### Q. Find the wrong number in the following series:

102, 101, 98, 93, 86, 74, 66, 53

(a) 66 (b) 74 (c) 70 (d) 54

(SSC CGL Question)

Explanation:

Pattern:

102 - 1 = 101

101 - 3 = 98

98 - 5 = 93

93 - 7 = 86

86 - 9 = **77**

77 - 11 = 66

66 - 13 = 53

Answer: (b)

## Number Series – Threat?

In some questions deciphering the underlying concept used may prove to be an uphill task. Hence, in exam you should adopt the following strategy, when it comes to attempting series questions (esp. in case of number series):

Read the series and see if any pattern is evident.

If not then do not waste more than 1 minute in any case. Move ahead…

Come back in the second round of the paper and give it another shot.