# Number Series

Generally, two kinds of series are asked in the examination.  One is based on numbers and the other based on alphabets.
In questions based on series, some numbers or alphabets are arranged in a particular sequence. You have to decipher that particular sequence of numbers or alphabets and on the basis of that deciphered sequence, find out the next number or alphabet of the series. Although there is no limit of logics which can be used to build a series, here are some important examples given which highlight the type of series asked in the examination.

### How to solve number series problems:

Step 1: Observer are there any familiar numbers in the given series.  Familiar numbers are primes numbers, perfect squares, cubes ... which are easy to identify.
Examples:
a. Prime number series: 2, 3, 5, 7, 11, 13, 17, . . .
b. Square series : 1, 4, 9, 16, 25, 36, 49, . . .
c. Cube series: 1, 8, 27, 64, 125, 216, 343, 512, 729, . . .
d. ${n^2} - 1$ series:  0, 3, 8, 15, 24,35, 48,
e. ${n^2} + n$ series: 2, 6, 12, 20, . . . .
Another Logic : The series is 1 x 2, 2 x 3, 3 x 4, 4 x 5, The next number is 5 x 6=30.

Step 2: Calculate the differences between the numbers.  Observe the pattern in the differences.  If the differences are growing rapidly it might be a square series, cube series, or multiplicative series.  If the numbers are growing slowly it is an addition or subtraction series.
If the differences are not having any pattern then
1.  It might be a double or triple series.  Here every alternate number or every 3rd number form a series
2.  It might be a sum or average series.  Here sum of two consecutive numbers gives 3rd number.  or average of first two numbers give next number
Example 1: 12, 9, 17, 13, 22, 17, 27, 21, 32, 25, 37, . . .
Answer: The above series is a mixed series.  12, 17, 22, 27, 32, 37, . . . form a series, 9, 13, 17, 21, 25, . . . form another series.  So after 37 we get 29.
Example 2: 720, 120, 24, 6, ?
Answer: 720/6 = 120, 120/5 = 24, 24/4 = 6, 6/3 = 2
Example 3: 32, 48, 72, 108, ?
48/32 = 3/2.  So Number x 3/2= next number. 32 x 3/2 = 48, 48 x 3/2 = 72, 72 x 3/2 = 108,  108 x 3/2 = 162.
Example 4: 2, 6, 18, 54, 162, ?
The numbers are multiplied by 3 to get next number. (162 x 3 = 486).

Step 3: If nothing works, then check image series, average series, etc.
Example: 34, 81, 72, 47, 74, 27, 18, ?
Answer :  Look at the middle two numbers. (47, 74).  They are images.  Similarly (72, 27), (81, 18) are images. So Image of 34 is 43.

### Number Series solved examples

1.  20, 17, 34, 31, 62, 59, ?
a.  95
b. 118
c. 110
d. 128
Explanation:
This series is based on the logic that previous  term –3, previous term ×2 and this pattern is repeated.  So next term is 59 × 2 = 118

2.  0, 7, 26, 63, 124, ?
a. 210
b. 215
c. 211
d. 224
Explanation: This series is based on cube of consecutive numbers minus one, starting from 1.  So next number is ${6^3} - 1$ = 215

3.  4, 8, 6, 5, 15, 10, 10, 20, 15, 17, 15, 16, 18, 14, ?
a. 16
b. 18
c. 20
d. 22
Explanation:
In every triplet, the third number is the average of the first two numbers. So next number is average of 18, 14 which is (14 + 18)/ 2 = 16

4.  2, 3, 4, 7, 13, 12, 12, 23, 20, 17, 33, 28, ?
a. 22
b. 29
c. 30
d. 32
Explanation:
It is a case of double alternate series.
2, –, –, 7, –, –, 12,–, –, 17, –, –, 22

5.  72, 9, 82, 10, 88, 16, 86, 14, 99, ?
a. 102
b. 22
c. 18
d. 16
Explanation:
72 = 7 + 2 = 9
82 = 8 + 2 = 10
Similarly, 99 = 9 + 9 = 18

6.  1, 2, 6, 15, 31, 56, 92, ?
a. 150
b. 128
c. 141
d. 149
Explanation: 2 - 1 = 1, 6 - 2 = 4, 15 - 6 = 9, 31 - 15 = 16, 56 - 31 = 25,
The differences are squares of natural numbers. So next difference is 36. So next number = 92 + 36 = 128

7.  4, 4, 12, 16, 36, 36, 108, ?
a. 64
b. 66
c. 82
d. 86
Explanation:
It is a mixture of two series.
4, 12, 36, 108 and 4, 16, 36 and 64.
The logic of 4, 12, 36, 108 is previous term × 3.
The logic of 4, 16, 36, is previous term +12, +20 +28, and so on. Therefore, the next term would be 64 at a difference of +28 from the last term. Alternatively, 4, 16, 36,. . . squares of even numbers. So next number would be 64.

8.  8, 9, 11, 15, 16, 18, ?
a. 21
b. 22
c. 23
d. 24
Explanation:
The difference between the two consecutive terms is following a pattern +1, +2 and +4. Therefore,  the next term  would be 18 + 4 = 22.

9.  20, 10, 30, 40, 70, 110, ?
a. 95
b. 180
c. 190
d. 195
Explanation:
Every number is sum of previous two terms. So next number = 70 + 110 = 180

10.  4, 5, 8, 17, 44, ?
a. 102
b. 104
c. 125
d. 110
Explanation:
Here differences are 1, 3,9, 27, . . . which are powers of 3. So next number = 44 + 81 = 125

11. 6, 14, 26, 42, 62, 86, ?
a. 114
b. 115
c. 116
d. 118
Explanation:
The difference between the terms is 8, 12, 16, 20, 24 and so on. Therefore, the next term would be at a difference of 28 from the last term. Hence, 114 will replace the question mark.

12.  5, 7, 9, 11, 14, 16, 20, 22, 27, 29, ?
a. 38
b. 31
c. 34
d. 35
Explanation:
Two series are mixed to form the given series.
I. 5, 9, 14, 20, 27 at a successive difference of 4, 5, 6, 7 ... and so on.
II. 7, 11, 16, 22, 29 at a successive difference of 4, 5, 6, 7 and so on. Hence, the next term would be
27 + 8 = 35.

13.  1, 1, 2, 4, 3, 9, 4, 16, 5, 25, 6, ?
a. 35
b. 42
c. 49
d. 36
Explanation:
Numbers at the even positions are the squares of numbers at the odd positions. So next number is 36

14.  12, 11, 10, 13, 18, 17, 14, 25, 24, 15, 32, 31, ?
a. 15
b. 34
c. 16
d. 38
Explanation:
Three series are mixed to form the given series.
I. 12, 13, 14, 15, 16, form one series
II. 11, 18, 25, 32, 39, form another series
III. 10, 17, 24, 31, 38 form another series.
So next number will be from the first series So 16.

### Number series Level - 2 Problems

1. 11, 31, 69, 131, 223, ?
a. 257
b. 351
c. 349
d. 231
Explanation:
The given series is, ${2^3} + 3$, ${3^3} + 4$, ${4^3} + 5$, ${5^3} + 6$, ${6^3} + 7$
Shortcut:  If the given numbers are in the format,  $f\left( x \right) = a{x^2} + bx + c$, then successive differences will give you constant.
20, 38,  62,  92,    128
18,  24,  30,  36
6,    6,    6
Note:  When less number of terms are given, successive differences method is difficult to employ.

2. 12, 15, 20, 27, 36, ?
a. 47
b. 59
c. 43
d. 49
Explanation:
The given series is ${1^2} + 11$, ${2^2} + 11$, ${3^2} + 11$, ${4^2} + 11$, ${5^2} + 11$, ${6^2} + 11$..
Shortcut:
Successive differences are
3, 5, 7, 9, 11
2,  2, 2, 2

3. 7, 5, 7, 17, 63, ?
a. 308
b. 302
c. 309
d. 409
Explanation:
The given series is
$7 \times 1 - 2 = 5$
$5 \times 2 - 3 = 7$
$7 \times 3 - 4 = 17$
$17 \times 4 - 5 = 63$
$63 \times 5 - 6 = 309$

4. 50, ?, 61, 89, 154, 280
a. 52
b. 51
c. 60
d. 62
Explanation:
The given series is
$50 + \left( {{1^3} + 1} \right) = 52$
$52 + \left( {{2^3} + 1} \right) = 61$
$61 + \left( {{3^3} + 1} \right) = 89$
$89 + \left( {{4^3} + 1} \right) = 154$
$154 + \left( {{5^3} + 1} \right) = 280$

5. 17, 19, 25, 37, ?, 87
a. 47
b. 37
c. 57
d. 67
$17 + \left( {1 \times 2} \right) = 19$
$19 + \left( {2 \times 3} \right) = 25$
$25 + \left( {3 \times 4} \right) = 37$
$37 + \left( {4 \times 5} \right) = 57$
$57 + \left( {5 \times 6} \right) = 87$