Series of Numbers: Rules

Series of Numbers

Perfect Square Numbers:

12 = 1                 22 = 4               32 = 9                 42 = 16               52 = 25

62 = 36               72 = 49             82 = 64               92 = 81              102 = 100

112 = 121          122 = 144         132 = 169           142 = 196           152 = 225

162 = 256          172 = 289         182 = 324           192 = 361           202 = 400

212 = 441          222 = 484         232 = 529           242 = 576           252 = 625

262 = 676          272 = 729         282 = 784           292 = 841           302 = 900

Examples

Series: 4   9   16   25   ?    49   64 

Solution

In this series, we can see the pattern is perfect squares of 2, 3, 4, 5, .............

So next perfect square will be      6 × 6 = 36

Series: 63   80   99   120   143   ?   195

Solution:
See    63    = 64 - 1     =   8 × 8 - 1
          80    = 81 - 4     =   9 × 9 - 1
          99    = 100 - 4    =  10 × 10 - 1
          120  = 121 - 4    =  11 × 11 - 1
          143  = 144 - 4    =   12 × 12 - 1

We can see that the pattern is    a  ×  a  - 1

So next number will be               13 × 13 - 1 = 169 - 1 = 168

Series: 125   104   85   68   ?    40

Solution

See    125 = 121 + 4 = 11 × 11 + 4

           104 = 100 + 4 = 10 × 10 + 4

             85 =   81 + 4 = 9 × 9 + 4

             68 =   64 + 4 = 8 × 8 + 4

We can see that the pattern is    a × a + 4

So next number will be                7 × 7 + 4 = 49 + 4 = 53

Perfect Cube Numbers:

13 = 1                 23 = 8               33 = 27               43 = 64             53 = 125

63 = 216             73 = 343           83 = 512             93 = 729         103 = 1000

113 = 1331       123 = 1728        133 = 2197        143 = 2744       153 = 3375

Examples

Series: 8   27   64   125   ?    343  512  

Solution:

In this series, we can see the pattern is perfect cubes of 2, 3, 4, 5, .............

So next perfect cube will be         6 × 6 × 6 = 216

Series: 63   124   215   342   ?   728

Solution:

See    63    = 64 - 1     =   4 × 4 × 4 - 1

        124    = 125 - 1    =   5 × 5 × 5 - 1

        215    = 216 - 1    =   6 × 6 × 6 - 1

        342    = 343 - 1    =   7 × 7 × 7 - 1

We can see that the pattern is    a × a × a - 1

So next number will be               8 × 8 × 8 - 1 = 512 - 1 = 511

Series: 1004   733   516   347  ?    129

Solution:

See    1004 = 1000 + 4  = 10 × 10 × 10  + 4

             733 =  729 + 4   = 9 × 9 × 9        + 4

             516 =  512 + 4   = 8 × 8 × 8        + 4

             347 =  343 + 4   = 7 × 7 × 7        + 4

We can see that the pattern is     a × a × a  + 4

So next number will be                 6 × 6 × 6 + 4 = 216 + 4 = 220

Geometric Progression

In these series, first we check for the multiplicand r by dividing preceding number from the previous number in the series. We know that GP series is formed in the following manner.

a   ar   ar2  ar3  ar4 ……………………..arn-2    arn-1  

Example:

Series: 1    2    4     8    16    32     64    ?    256     512

Solution:

Divide 2 / 1 = 2,     4 / 2 = 2,     8 / 4 = 2,    .... and so on.....   So we can say that,

This is a series with a = 1 and r = 2

So series will be 

1 × 2 = 2,       2 × 2 = 4,      4 × 2 = 8,        8 × 2 = 16,         16 × 2 = 32,

32 × 2 = 64,       64 × 2 = 128

So the missing number will be 128

The same can be found using formula: ar6 - 1 = ar7 = 1 × (2)7 = 128

Series: 64    32    16    8    4    2    1     ?

Solution:

Divide 32 / 64 = 1/ 2,    16 / 32 = 1/ 2,    8 / 16 = 1 / 2,    .... and so on.....  So we can say that,

This is a series with a = 64 and r = 1/2

So series will be 

64 × 1/2 = 32,     32 × 1/2 = 16,      16 × 1/2 = 8,       8 × 1/2 = 4,      4 × 1/2 = 2,

2 × 1/2 = 1,       1 × 1/2 = 1/2 = 0.5

So the missing number will be 1/2  or  0.5

The same can be found using formula: ar6 - 1 = ar7 = 64 × (1/2)7 = 64 / 128 = 1 / 2 = 0.5   

Series: 3    4    6     10    18    ?     66    130    258

Solution:

This is a series with a = 3 and r = 2 with a difference of 2 as follows

3 × 2 - 2 = 6 - 2 = 4,                4 × 2 - 2 = 8 - 2 = 6,              6 × 2 - 2 = 12 - 2 = 10,

10 × 2 - 2 = 20 - 2 = 18          18 × 2 - 2 = 36 - 2 = 34          34 × 2 - 2 = 68 - 2 = 66

So the missing number will be 34

Arithmetic Progression:

In these series, first we check for the difference d by subtracting previous number from the preceding number in the series. We know that AP series is formed in the following manner.

a        a + d        a + 2d          a + 3d …………….. a + (n-2)d         a + (n-1)d

Example:

Series: 4      9      14    19    ?     29    34     39

Solution:

Subtracting  9 - 4 = 5,      14 - 9 = 5,      19 - 14 = 5,    .... and so on.....   So we can say that,

This is a series with a = 4 and d = 5

4 + 5 = 9,      9 + 5 = 14,      14 + 5 = 19,     19 + 5 = 24,      24 + 5 = 29, ....

So the missing number will be 24

The same can be found using formula: a + (n - 1)d = 4 + (5 - 1)× 5 = 4 + 20 = 24

Series:    37   30     23    16    9      2      ?     -12 

Solution:

Subtracting  30 - 37 = - 7,  23 - 30 = - 7,  16 - 23 = -7, .... and so on.... So we can say that,

This is a series with a = 37 and d = - 7

37 - 7 = 30,      30 - 7 = 23,     23 - 7 = 16,      16 - 7 = 9,     9 - 7 = 2,      2 - 7 = - 5

So the missing number will be (- 5)

The same can be found using formula: a + (n - 1)d = 37 + (7 - 1)× (- 7) = 37 - 42 = - 5

Arithmetic Progression with variable Difference (d):

In these series, first we check for the difference 'd' by dividing preceding number from the previous number in the series. We know that GP series is formed in the following manner.

We will see that the values of 'd' in each calculation is different but having some definite pattern. we can understood it using some examples.

Example:

Series: 2      5      9    14    20     27    ?

Solution:

Subtracting  d = 5 - 2 = 3,      d = 9 - 5 = 4,      14 - 9 = 5,    20 - 14 = 6,    27 - 20 = 7

See the sequence of d = 3       3 + 1 = 4       4 + 1 = 5       5 + 1 = 6       6 + 1 = 7

So next d will be     7 + 1 = 8

So next value in the series will be   27 + 8 = 35

Geometric Progression with variable Multiplicand (r):

In these series, first we check for the multiplicand r by dividing preceding number from the previous number in the series. We know that GP series is formed in the following manner.

We will see that the values of 'r' in each calculation is different but having some definite pattern. we can understood it using some examples.

Example:

Series:     2     6    24    120    720    ?

Solution:

Divide:   r = 6 / 2 = 3,     r = 24 / 6 = 4,    r = 120 / 24 = 5,     r = 720 / 120 = 6

See the sequence of r = 3       3 + 1 = 4       4 + 1 = 5       5 + 1 = 6   

So next r will be     6 + 1 = 7

So next value in the series will be   720 × 7 = 5040

Prime numbers:

These are the numbers which are either divisible by 1 or itself, means it has only and necessarily two factors.

1 is not a prime number because it has only one factor that is 1.

2 is only even prime number which has two factors that is 1 and 2 itself.

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199

Series: 5    7    11    13    17    ?

Solution:

We can see that this is the series of prime numbers. So next number will be 19

Series: 3    7    13    19    29    ?

Solution:

We can see that this is the series of alternate prime numbers. 

3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37

Taking alternate numbers we get     3    7    13    19    29    37

So next number will be 37

Series: 26    34    38    46    58    ?

Solution:

We can see that there is neither AP nor GP or any other theory in this series. but all numbers are even numbers.

Divide all numbers by 2, and we get

26 / 2 = 13,     34 / 2 = 17,     38 / 2 = 19,     46 / 2 = 23,     58 / 2 = 29

Here we can see the hidden prime number series 13, 17, 19, 23, 29

Next prime number will be 31

So next number in the series will be       31 × 2 = 62