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Number Series  A Series is a sequence of numbers obtained by some particular predefined rule and applying that predefined rule we...

# 01. Verbal Reasoning for Placement Papers --- NUMBER SERIES - Theory and Concept

Number Series
A Series is a sequence of numbers obtained by some particular predefined rule and applying that predefined rule we can write the succeeding terms of the series.
For Example: 1, 3, 5, 7, 9 … is a series and 1,3,5,7 and 9 individually are called terms of the series. The dots after 9 in the series mean that the series continues. We can say that.
(i.. The above example is a series of odd numbers or
(ii.. It is a series starting with 1 with succeeding terms exceeding the previous term by 2.
So, that next term after 9 will be 11.
Important number series:
Pure Series: In this type of number series, the number itself obeys certain order so that the number (term. of the series can be found out.
The number it may be:
(i) Prime number: It is a whole number greater than 1, which is divisible by itself and 1.
(ii) Even Number: A number which is divisible by 2 is called even number.
(iii) Odd Number: A number which is not divisible by 2 odd number
Ex: 1, 3, 5, 7, 9, 11 etc)
(iv) Perfect square: A Whole number whose square root is also a whole number is called a perfect square.
Ex: 1, 4, 9, 16, 25, 36…etc
(v) Perfect cube: A whole number whose cube root is also a whole number is called a perfect cube.
Ex: 1, 8, 27, 64, 125 etc
(vi) Square & cube of prime number
Ex: 4, 9, 25, 49, 121
Ex: 8, 27, 125, 343, 1331.

Difference Series (Arithmetic Series)

In This type of series, each successive number in the sequence of numbers is obtained by adding or subtracting a fixed number to or from the previous number.
Change in order for the Difference Series:
(i)          Difference between consecutive numbers is same:
2, 5, 8, 11, 14                  190 185 180 175 170
+3 +3 +3 +3                         -5    -5    -5    -5
(ii) Differences    between consecutive numbers are in Arithmetic progression.
10, 14, 22, 34, 50                           30, 21, 14, 9, 6
+4  +8 +12 + 16                                -9   -7  -5  -3
(iii)      Differences between consecutive numbers are prime numbers.
2, 4, 7, 12, 19, 30         29, 22, 15, 4
+2 +3 +5 +7 +11              -5  -7  -11
(iv)      Differences between consecutive numbers are in G. P.
1   2   5  14  41  124            30, 14, 6, 2, 0
+1 +3 +9 +27 +81                -16, -8, -4, -2
(v)   Difference between consecutive numbers is a perfect square numbers.
6   10   19    35              70,  54, 45,  41,  40
+22   +32   +42                    -42   -32   -22   -12
(vi)      Difference between consecutive numbers is a perfect cube number.
5,  13,  40,  104              120  56  29  21
+23  +33  +4              -43  -33  -23
(vii)    Differences between consecutive numbers are a multiple of a number.
5, 12, 26, 37, 65                        131, 83, 47, 23, 11
+7,  +14,+21, +26                       -48,  -36,  -24, -12
Multiple of 7                Multiple of 12

Ratio Series (Geometric Series).:A Ratio series is one in which each successive number is obtained by multiplying or dividing a fixed number by the previous number.
Change in order for the ratio series:
(i)          Ratios between consecutive numbers are  same.
2,  6,  18,  54,  162                        64,  32,  16,  8,  4
×3  ×3  ×3  ×3                ÷2  ÷2  ÷2  ÷2
(ii)        Ratios between consecutive numbers are in A. P.
3,  6,  18,  72,  360                        860,  192,  48,  16,  8
×2  ×3  ×4  ×5                  ÷5     ÷4     ÷3   ÷2
(iii)      Ratio between consecutive numbers is a prime number.
5,  10,  30,  150,  1050,  11550
×2   ×3   ×5     ×7       ×11
770,  70,  10, 2
÷11  ÷7  ÷5
(iv)      Ratios between consecutive numbers are in G.P.
1, 2, 8, 64, 1024  2                      048,  128,  16,  4,  2
×2 x4 ×8 ×16                          ÷16  ÷8  ÷4  ÷2
(v)        Ratio between consecutive numbers is a perfect square number.
3,  12,  192,  6912                                   44100, 900, 36, 4
×22  ×42  ×62                              ÷72  ÷52  ÷32
(vi)      Ratio between consecutive numbers is a perfect cube number.
2, 2, 16, 432, 27648      69120, 1080, 40, 5, 5
×13 ×23  ×33  ×43            ÷43   ÷33  ÷23  ÷13
(vii)    Ratio between consecutive number is the multiple of a number
4,   8,  32,  252,  4096          972, 108, 18, 6
×2  ×4  ×8  ×16                        ÷9  ÷6  ÷3

Mixed Series:

It consists of two or more series combined into a single series. The series so formed are termed as combination series or mixed series.
Ex: 2, 3, 7, 9, 12, 27, 17, 22, 243.
In this series, the terms at odd places of the series form an arithmetic series while the even terms of the series form an independent geometric series.
Odd terms: 2, 7, 12, 17, 22 is a difference series with common difference 5.
Even terms: 3, 9, 27, 81, 243.. is a ratio series with common ratio2 3.

Miscellaneous Series:

1)      2+1 Series:
Example: 2, 5, 10, 17, 26, 37, __
The series is 12+1, 22+1, 32+1 etc)
The next number is 72+1=50.

2)      2-1 Series:
Example: 0, 3, 8, 15, 24, 35, 46, 63, __
The series is 12-1, 22-1, 32-1, 42-1, 52-1, 62-1 etc)
The next number is 82-1=63.

3)      2+n Series:
Example: 6, 12, 20, 30, 42, __
The series is 22+2, 32+3, 42+4, 52+5, 62+6.
The next number is 72+7=56.

4)      2-2 Series:
Example: 2, 6, 12, 20, 30, __
The series is 22-2, 32-3,
so next number is 72-7=42

5)      n3+1 Series:
Example: 28, 65, 126, 217, 344, __
The series is 33+1, 43+1,… 73+1.
The next term is 83+1=513.

6)      3-1 Series:
Example: 26, 63, 124, 215, ___
The Series is 33-1, 43-1, 53-1, 63-1.
The next number is 73-1=342.

7)      3+n Series:
Example: 2, 10, 30, 68, 130, __
The Series is 13+1, 23+2, 33+3 etc)
The next number is 63+6=222.
8)      3-n Series:
Example: 24, 60, 120, 210, __
The Series is 33-3, 43-4, 53-5, 63-6.
The next number is 73-7=336.

9)      p2-p where p is prime number .
Example: 2, 6, 20, 42, 110, __
The series is 222, 32-3, 52-5, 72-7, 112-11
so next number is 132-13=156.

10)  2+p where p is prime number.
Ex: 6, 12, 30, 56, 132, ___
The series is 22+2, 32+3, 52+5, 72+7, 112+11.
Next number will be 132+13=182.

11)  3-n2 Series.
Example: 4, 18, 48, 100, __
The series is 23-22, 33-32, 43-42, 53-52.
The next number is 63-62=180.
12)  3+n2 series.
Example: 12, 36, 80, 150, __
The series is 13+12, 23+22, 33+32, 43+42,53+52.
The next number is 63+62=252.

13)  x+y  Series:
Example: 48, 12, 76, 13, 54, 9, 32__ 4+8=12
7+6=13  5+4=9 so 3+2=5.

14)  xy Series:
Ex:-12, 2, 22, 4, 33, 9, 44, 16, 27, __
1×2=2, 2×2=4,  3×3=9, 4×4=16 so 5×7=35.

15)  Interchange Series:
Ex: 34, 84, 72, 47, 74, 27, 48, __
47,74,.  72,27,    84,48. are inter changed)
So34, 43. is another pair.

16)  Palindrome Series :
Example : 121,242,343, 515,686
The next can be any palindrome number

17)  n n 2  Series :
Example: 11 ,24, 39, 416,525
The series is 1 12,,2 22,  3 32, 4 42, 5 52
The next number is 6 62. So next number is 636.

18)  nnseries :
Ex: 11, 28, 327 ,464, ..
The next number will be 5125

19)  n2nseries :
Example: 11, 48, 927,1664,..
So the next number will be 25125

20)  Pair of Consecutive Prime Numbers :
Ex: 23, 35,57,711,1113
The next number is 1317

21)  $\frac{n}{n^{2}}$ series :
Example: $1,\,\frac{2}{4} ,\, \frac{3}{9} ,\,\frac{4}{16},\,\frac{5}{25}$ ..
The next number will be $\frac{6}{36}$

22)  $\frac{n^2}{n^{3}}$
series :
Example : $1,\,\frac{4}{8} ,\, \frac{9}{27} ,\,\frac{16}{64},\,\frac{25}{125}$
The next number is $\frac{36}{216}$

23)  $\frac{n}{n^{2}+1}$
Example :$\frac{1}{2} ,\, \frac{2}{5} ,\,\frac{3}{10},\,\frac{4}{17} \frac{5}{25}$
The next term is$\frac{6}{37}$.

24)  $\frac{1}{n^{x}+1}$       where x is any integer
Example : $\frac{1}{2} , \frac{1}{3} ,\, \frac{1}{5} ,\,\frac{1}{9},\,\frac{1}{17}$

The terms in the given series are

$\frac{1}{2^{0}+1} ,\,\frac{1}{2^{1}+1}, \, \frac{1}{2^{2}+1}$ and so on.

The next term will be $\frac{1}{2^{5}+1}$ i.e, $\frac{1}{33}$

25)Divisibility by a certain number .

Example : 132 , 165,198, 220,

a) 242        b) 235  c) 284  d)256

Option b is right choice, because all the numbers in the given series are divisible by 11

26)  Operations on the digits of the numbers

Example: 124 , 2122, 4112, __

a) 12114     b) 3612 c) 3421        d) 34512

Option a is correct choice because the products of the digits of all other numbers have the same value 8..

Ex2 : 236,428,122,4312,

a) 8322    b) 9327 c) 5678              d) 8902

Option b is the right choice because the product of the first and second digit of the given number is its remaining right digits.

27)  Positional Uniformity

Ex:137,238,11356,939,

a) 23454 b) 383 c) 24367 d) 8793

Option c is right choice because, all numbers having 3 as their middle digit