## 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.

**E x**

**ample**- 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

#### n^{2} 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 n^{2} series.

**Example**- Find out the next term in the series 4, 9, 25, 49, 121, ….**Solution**:- 2^{2} = 4

3^{2} = 9

5^{2} = 25

7^{2} = 49

11^{2} = 121

This is a series of square of consecutive prime number.

∴ Next term = 13^{2} = 169.

#### ( n^{2} + 1 ) Series :-

In this type of series, each term is a sum of a square term and 1, then this series is called ( n^{2} + 1 ) series.

**Example**- Find out the missing term in the series 122, 101, 82, 65, 50, **?**, 26.**Solution**:- Series pattern, 11^{2} + 1 = 122

10^{2} + 1 = 101

9^{2} + 1 = 82

8^{2} + 1 = 65

7^{2} + 1 = 50

6^{2} + 1 = 37

5^{2} + 1 = 26

∴ Required number = 6^{2} + 1 = 37

#### ( n^{2} - 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 ( n^{2} - 1 ) series.

**Example**-Find out the missing term in the series 48, 63, 80, 99, 120,….**Solution**:- Series pattern

7^{2} - 1 = 48

8^{2} - 1 = 63

9^{2} - 1 = 80

10^{2} - 1 = 99

11^{2} - 1 = 120

∴ Required number = 12^{2} - 1 = 143

#### ( n^{2} + 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 ( n^{2} + n ) series.

**Ex**- Find out the next term in the series 110, 90, 72, 56, 42, ….**Solution**:- Series pattern

10^{2} + 10 = 110

9^{2} + 9 = 90

8^{2} + 8 = 72

7^{2} + 7 = 56

6^{2} + 6 = 42

∴ Required number = 5^{2} + 5 = 30

#### ( n^{2} - 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 ( n^{2} - n ) series.

**Example**- Find out the next term in the series 0, 2, 6, 12, 20, 30, ….**Solution**:- Series pattern

1^{2} - 1 = 0

2^{2} - 2 = 2

3^{2} - 3 = 6

4^{2} - 4 = 12

5^{2} - 5 = 20

6^{2} - 6 = 30

∴ Required number = 7^{2} - 7 = 42

#### n^{3} 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 n^{3} series.

**Example**- Find out the missing term in the series 216, 125, 64, 27, **?**, 1**Solution**:- Series pattern

6^{3} = 216

5^{3} = 125

4^{3} = 64

3^{3} = 27

2^{3} = 8

1^{3} = 1

∴ Required number = 2^{3} = 8

#### ( n^{3} + 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 ( n^{3} + 1 ) series.

**Example-** Find out the missing term in the series 2, 9, 28, 65, **?**, 217**Solution**:- Series pattern

1^{3} + 1 = 2

2^{3} + 1 = 9

3^{3} + 1 = 28

4^{3} + 1 = 65

5^{3} + 1 = 126

6^{3} + 1 = 217

∴ Required number = 5^{3} + 1 = 126

#### ( n^{3} - 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 ( n^{3} - 1 ) Series.

**Example**- Find out the missing term in the series 215, 124, 63, 26, **?**, 0**Solution**:- Series pattern

6^{3} - 1 = 215

5^{3} - 1 = 124

4^{3} - 1 = 63

3^{3} - 1 = 26

2^{3} - 1 = 7

1^{3} - 1 = 0

∴ Required number = 2^{3} - 1 = 7

#### ( n^{3} + 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 ( n^{3} + n ) series.

**Example**- Find out the missing term in the series 222, 130, 68, 30, **?**, 2**Solution**:- Series pattern

6^{3} + 6 = 222

5^{3} + 5 = 130

4^{3} + 4 = 68

3^{3} + 3 = 30

2^{3} + 2 = 10

1^{3} + 1 = 2

∴ Required number = 2^{3} + 2 = 10

#### ( n^{3} - n ) Series :-

In this type of series each term is obtained by subtracting the number from its cube, and such series is called ( n^{3} - n ) series.

**Example**- Find out the missing term in the series 210, 120, 60, 24, **?**, 0.**Solution**:- Series pattern

6^{3} - 6 = 210

5^{3} - 5 = 120

4^{3} - 4 = 60

3^{3} - 3 = 24

2^{3} - 2 = 6

1^{3} - 1 = 0

∴ Required number = 2^{3} - 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 a_{1} = a

second term a_{2} = a + d

third term a_{3} = a + 2d

forth term a_{4} = 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

a_{12} = 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, ar^{2}, ar^{2},…. is known as a geometric progression with first term a and common ratio r.

Common ratio = | next term | = | ar | = | ar^{2} |
= | ar^{3} |
...... = r |

previous term | a | ar | ar^{2} |

second term = ar

third term = ar

^{2}

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 |

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.