A number series is a pattern which is written from left to right in a sequence.
To solve the questions on series, find the pattern that is followed in the series between the consecutive terms.

Types of Series :-

Prime number Series :-

A number which has only two factors which are 1 and the number itself is called prime number. The series formed by prime numbers is called prime number series.

Example- Find out the next term in the series 2, 3, 5, 7,…..
Solution:- Given series is a consecutive prime number series. Therefor the next term will be 11.

Example- Find out the missing term in the series 3, 5, 7, ?, 13, 17, 19, 23
Solution:- Given series is a consecutive prime number series. Therefore, missing number will be prime number after 7 which is 11.

Addition Series :-

In this type of series next term is obtained by adding a specific number to the previous term. Such series is called addition series.

Example- Find out the missing term in the series 7, 13, 19, 25, ?, 37
Solution:- 7 + 6 = 13
13 + 6 = 19
19 + 6 = 25
25 + 6 = 31
31 + 6 = 37
Here, every next term is obtained by adding 6 to previous term.
∴ Missing number = 25 + 6 = 31

Difference Series :-

In this type of series next term is obtained by subtracting a specific number from the previous term.

Example- Find out the next term in the series 420, 408, 396, 384, 372,….
Solution:- 420 - 12 = 408
408 - 12 = 396
396 - 12 = 384
384 - 12 = 372
Here, every next term is obtained by subtracting 12 from previous term.
∴ Required term = 372 - 12 = 360

Multiple Series :-

In this type of series, each term of the series is obtained by multiplying a number with the previous term. This type of series is called multiplication series.

Example- Find out the missing term in the series 6, 12, 24, ?, 96, 192.
Solution:- 6 × 2 = 12
12 × 2 = 24
24 × 2 = 48
48 × 2 = 96
96 × 2 = 192
Here, every next term is double of the previous term.
∴ Required term = 24 × 2 = 48

Division Series :-

In this type of series, each term is obtained by dividing the previous term by a number.

Example- Find out the next term in the series 10000, 2000, 400, 80, 16, 3.2,…
Solution:- 10000 ÷ 5 = 2000
2000 ÷ 5 = 400
400 ÷ 5 = 80
80 ÷ 5 = 16
16 ÷ 5 = 3.2
Here every next term is obtained by dividing by 5
∴ Next term = 3.2 ÷ 5 = 0.64

n² Series :-

If a number is multiplied by itself, it is called square of the number and the series formed by square of numbers is called n² series.

Example- Find out the next term in the series 4, 9, 25, 49, 121, ….
Solution:- 2² = 4
3² = 9
5² = 25
7² = 49
11² = 121
This is a series of square of consecutive prime number.
∴ Next term = 13² = 169.

( n² + 1 ) Series :-

In this type of series, each term is a sum of a square term and 1, then this series is called ( n² + 1 ) series.

Example- Find out the missing term in the series 122, 101, 82, 65, 50, ?, 26.
Solution:- Series pattern, 11² + 1 = 122
10² + 1 = 101
9² + 1 = 82
8² + 1 = 65
7² + 1 = 50
6² + 1 = 37
5² + 1 = 26
∴ Required number = 6² + 1 = 37

( n² - 1 ) Series :-

In this type of series, each term is obtained by subtracting 1 from the square of a number, then such series is known as ( n² - 1 ) series.

Example-Find out the missing term in the series 48, 63, 80, 99, 120,….
Solution:- Series pattern
7² - 1 = 48
8² - 1 = 63
9² - 1 = 80
10² - 1 = 99
11² - 1 = 120
∴ Required number = 12² - 1 = 143

( n² + n ) Series :-

In this type of series each term is obtained by sum of a number with square of that number. Such series is called ( n² + n ) series.

Ex- Find out the next term in the series 110, 90, 72, 56, 42, ….
Solution:- Series pattern
10² + 10 = 110
9² + 9 = 90
8² + 8 = 72
7² + 7 = 56
6² + 6 = 42
∴ Required number = 5² + 5 = 30

( n² - n ) Series :-

In this type of series each term is obtained by subtracting a number from square of that number, and the series is known as ( n² - n ) series.

Example- Find out the next term in the series 0, 2, 6, 12, 20, 30, ….
Solution:- Series pattern
1² - 1 = 0
2² - 2 = 2
3² - 3 = 6
4² - 4 = 12
5² - 5 = 20
6² - 6 = 30
∴ Required number = 7² - 7 = 42

n³ Series :-

If a number is multiplied with itself twice, then the resulting number is called the cube of a number and series which consists of cube of different numbers is called as n³ series.

Example- Find out the missing term in the series 216, 125, 64, 27, ?, 1
Solution:- Series pattern
6³ = 216
5³ = 125
4³ = 64
3³ = 27
2³ = 8
1³ = 1
∴ Required number = 2³ = 8

( n³ + 1 ) Series :-

In this type of series, each term is obtained by sum of cube of a number and 1, and such a series is called ( n³ + 1 ) series.

Example- Find out the missing term in the series 2, 9, 28, 65, ?, 217
Solution:- Series pattern
1³ + 1 = 2
2³ + 1 = 9
3³ + 1 = 28
4³ + 1 = 65
5³ + 1 = 126
6³ + 1 = 217
∴ Required number = 5³ + 1 = 126

( n³ - 1 ) Series :-

In this type of series, each term is obtained by subtracting 1 from the cube of a number, and the series is called ( n³ - 1 ) Series.

Example- Find out the missing term in the series 215, 124, 63, 26, ?, 0
Solution:- Series pattern
6³ - 1 = 215
5³ - 1 = 124
4³ - 1 = 63
3³ - 1 = 26
2³ - 1 = 7
1³ - 1 = 0
∴ Required number = 2³ - 1 = 7

( n³ + n ) Series

In this type of series, each term is obtained by sum of a number with its cube, then the series is known as ( n³ + n ) series.

Example- Find out the missing term in the series 222, 130, 68, 30, ?, 2
Solution:- Series pattern
6³ + 6 = 222
5³ + 5 = 130
4³ + 4 = 68
3³ + 3 = 30
2³ + 2 = 10
1³ + 1 = 2
∴ Required number = 2³ + 2 = 10

( n³ - n ) Series :-

In this type of series each term is obtained by subtracting the number from its cube, and such series is called ( n³ - n ) series.

Example- Find out the missing term in the series 210, 120, 60, 24, ?, 0.
Solution:- Series pattern
6³ - 6 = 210
5³ - 5 = 120
4³ - 4 = 60
3³ - 3 = 24
2³ - 2 = 6
1³ - 1 = 0
∴ Required number = 2³ - 2 = 6

Alternating Series :-

In this type of series, successive terms increase and decrease alternately.
The key aspect of an alternating series is that it is a combination of two different series. On these two different series, two different operations are performed on successive terms alternately.

Example- Find out the next term in the series 7, 9, 12, 16, 17, 23, 22,….
Solution:- Series pattern
7 + 5 = 12
12 + 5 = 17
17 + 5 = 22
Here, every next term is obtained by adding 5 to the previous term.
9 + 7 = 16
16 + 7 = 23
23 + 7 = 30
Here, every next term is obtained by adding 7 to the previous term.
∴ Required term = 23 + 7 = 30

Arithmetic Progression :-

The progression of the form a, a + d, a + 2d, a + 3d,….. is called arithmetic progression with first term a and common difference d.
Common difference = second term - first term = third term - second term = …
first term a₁ = a
second term a₂ = a + d
third term a₃ = a + 2d
forth term a₄ = a + 3d
nth term a_n = a + ( n - 1 ) d
∴ a_n = a + ( n - 1 ) d

Example- In series 105, 112, 119,……. , what will be the 12th term?
Solution:- 112 - 105 = 119 - 112 = 7
The given series is in the form of Arithmetic Progression, since the common difference, d is same.
Here, a = 105, d = 7
∴ a_n = a + ( n - 1 ) d
a₁₂ = 105 + ( 12 - 1 ) 7
= 105 + 77
= 182

Geometric Progression :-

A series in which ratio between two successive terms is same, is called geometric progression.
The progression of the form a, ar, ar², ar²,…. is known as a geometric progression with first term a and common ratio r.

Ex- In series 12, 24, 48,……. , what will be the 11th term?

To Find the Wrong term :-

A series in which one of term is required to detect the pattern of series and find that wrong term.

Ex- Find the wrong number in the given series 13, 25, 40, 57, 79, 103, 130.
Solution:- Series pattern
13 + 12 = 25
25 + 15 = 40
40 + 18 = 58
58 + 21 = 79
79 + 24 = 103
103 + 27 = 130
Clearly, 57 is the wrong number and will be replaced by 58.