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 +43 -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) n 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) n 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) n 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) n 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) n 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) n 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) n 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) p 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) n 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) n 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) nn3 series :
Ex: 11, 28, 327 ,464, ..
The next number will be
5125
19) n2n3 series :
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) $\frac{n^2}{n^{3}}$
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