## Sequences and Series

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 an = a + ( n - 1 ) d
∴ an = 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
∴ an = 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.

 Common ratio = next term = ar = ar2 = ar3 ...... = r previous term a ar ar2

First term = a
second term = ar
third term = ar2
nth term = ar ( n - 1 )

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

 Solution:- Common ratio = 24 = 48 = ...... = 2 12 24

The given series is in the form of Geometric Progression, since the common ratio is same.
Here, a = 12, r = 2
∴ 11th term = ar( n - 1 )
= 12 × ( 2 )( 11 - 1 )
= 12 × ( 2 )( 11 - 1 )
= 12 × ( 2 )10
= 12 × 1024
Hence , 11th term = 12288

#### 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.