## Number Series

A number series is a pattern which is written from left to right or right to left in a sequence. To solve the questions on series, find the pattern that is followed in the series between the consecutive terms. In this chapter we will discuss different type of questions based on number series. In these type of questions candidates have to follow a definite pattern throughout the series. The tasks for the candidate is to identify the pattern and complete the given series with the best alternative or to find the wrong term in the series.

#### DIFFERENT TYPES OF NUMBER SERIES :-

Prime number Series :-
A number which have two factors, factors are 1 and number itself is called prime number. The series formed by using prime number is called prime number series.

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. Therefor missing number will be next prime number of 7 which is 11.

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

Example: 3, 4, 6, 9, 13, 18, __,
Solution: 3 + 1 = 4
4 + 2 = 6
6 + 3 = 9
9 + 4 = 13
13 + 5 = 18
18 + 6 = 24
Each number is increasing by consecutive number
∴ Required number = 18 + 6 = 24

Multiple Series :-
In this type of series each term of a series is obtained by multiplying a number with the previous term, then the series is called multiplication series.

Example: 8 , 28 , 116 , 584 , ?
(a) 1752 (b) 3502 (c) 3504 (d) 3508 (e) None of these
Solution: The given series is in the pattern: × 3 + 4, × 4 + 4, × 5 + 4, …..
(8 × 3) + 4 = 28
(28 × 4) + 4 = 116
(116 × 5) + 4 = 584
(584 × 6) + 4 = 3508
So, the missing term is 584 × 6 + 4 = 3508.
Ans: Option (d)

Example: 3 , 4 , 10 , 33 , 136 , 685 , ? (a) 3430 (b) 4802 (c) 5145 (d) 4116 (e) 5488
Solution: In this particular Series, the pattern is
3 × 1 + 1 = 4
4 × 2 + 2 = 10
10 × 3 + 3 = 33
33 × 4 + 4 = 136
136 × 5 + 5 = 685
So, the messing term is 685 × 6 + 6 = 4116.
Ans: option (d)

Square, Square root and cube and cube root series :-

Example: 1, 9, 25, 49,?, 121
(a) 64 (b) 81 (c) 91 (d) 100 (e) None of these
Solution: The given series consists of squares of consecutive odd numbers i.e 12, 32, 52, 72,…..
So, the missing term = 92 = 81
Ans: Option (b)

Triangular Number :-
When any number series is in the form a, a + (a + 1), a + (a + 1) + (a + 2), a + (a + 1) + (a + 2) + (a + 3), ……, nth term of the series be

 n ( n + 1 ) 2

Ex- 1, 3, 6, 4, 15, 21, …… find the 15th term of the series?

 Solution: 15th term = n ( n + 1 ) 2
 15th term = 15 ( 15 + 1 ) 2
= 15 x 8 = 120

Product of the digits and summation with any constant, gives next number of the series .

Example: 69, 55, 26, 13, 5 [Find the Wrong term of the Series]
(a) 69 (b) 55 (c) 26 (d) 13 (e) 5
Solution: We found that the series contains each term is one more than the product of the digits of the preceding term.
Thus (6 × 9) + 1 = 55, (5 × 5) + 1 = 26, (2 × 6) + 1 = 13, (1 × 3) + 1 = 4.
Ans: Option (e)

Difference Between the adjacent numbers increases by a constant number

Example: 2, 7, 17, 32, ?
(a) 42 (b) 52 (c) 47 (d) 27 (e) 62
Solution: In this particular series the gap between the numbers are 5, 10, 15, …
So the next difference must be 20. Than the next number be 52.
Ans: option (b)

Percentage increasing and decreasing series :-

Example: 105, 94.5, 103.95, 93.56, ?
(a) 102.91 (b) 102.98 (c) 102.95 (d) 103. 91 (e) none of these
Solution: In this particular Series the numbers are arranged on percentage increase and decrease pattern. As first there is 10% decrease and than 10% increase..i.e
105 – 10% of 105 = 94.5
94.5 + 10% of 94.5 = 103.95 …. and so on
Hence option (a) 102.91 is the very next number of the series.

Division or Multiplication with increasing and decreasing numbers :-
Example: 14: 51975, 9450, 2100, 600, 240, 160?
(a) 80 (b) 120 (c) 320 (d) 240 (e) 300
Solution:-

Example: 16, 12, 18, 40.5, 121.5, 455.625?
(a) 2050.1125 (b) 2050.2125 (c) 2050.3125 (d) 2050.4125 (e) 2050.5125
Solution: The pattern of the above Series: 16 × 0.75 = 12
12 × 1.5 (i.e 0.75 + 0.75 = 1.5) = 18
18 × 2.25 (i.e 1.5 + 0.75 = 2.25) = 40.5
40.5 × 3 (i.e 2.25 + 0.75) = 121.5, ……,
455.625 × 4.50 = 2050.3125.
Ans: Option (c)

Example: 4,18, 48, 100, 180, 294? (a) 448 (b) 424 (c) 436 (d) 460 (e) 412
Solution: The pattern is, here 3rd order difference is constant.