# Number Series

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# Number Series

### Introduction

Number Series is the arrangement of numbers in a certain order where some numbers are wrongly kept or some numbers are missing from that series. So accurate series are to be found. Number Series in competitive exams are divided into two. One is missing series and the other is wrong series. A number series is given in which a number is wrongly placed is the wrong series. Candidates are asked to identify that particular wrong number. A number series in which a specific number is missing is the missing series. Candidates have to identify the missing number.

### Levels

Missing number series will be based on easy, moderate and difficult type questions.
1. Easy- Under easy type, there are two scenarios.
Scenario 1: One is based on addition/subtraction by common difference
Scenario 2: Other is based on multiplication/division-Here a certain no. is multiplied or divided to the previous number to get the next number.
2. Moderate - a) we need to find out common difference and b)pattern of common difference (addition, subtraction, multiplication, division).
3. Difficult - Here common difference does not follow the pattern.
Wrong number series - In this series all the numbers will be followed by one rule except the one which is wrong.

### Methods

Perfect Square Series: This Types of Series are based on square of a number which is in same order and one square number is missing in that given series.
Example 1: 841, ?, 2401, 3481, 4761
Solution:
[latex]29^{2}[/latex], [latex]39^{2}[/latex], [latex]49^{2}[/latex], [latex]59^{2}[/latex], [latex]69^{2}[/latex]

Example 2: 1, 9, 25, ?, 81, 121
Solution:
[latex]1^{2}[/latex], [latex]3^{2}[/latex], [latex]5^{2}[/latex], [latex]7^{2}[/latex], [latex]9^{2}[/latex], [latex]11^{2}[/latex]

Example 3: 289, 225, 169, ?, 81
Solution:
[latex]17^{2}[/latex], [latex]15^{2}[/latex], [latex]13^{2}[/latex], [latex]11^{2}[/latex], [latex]9^{2}[/latex]

Perfect Cube Series: This Types of Series are based on cube of a number which is in same order and one cube number is missing in that given series.
Example 1: 3375, ?, 24389, 46656, 79507
Solution:
[latex]15^{3}[/latex], [latex]22^{3}[/latex], [latex]29^{3}[/latex], [latex]36^{3}[/latex], [latex]43^{3}[/latex]
(Each cube digit added with seven to become next cube number)

Example 2: 729, 6859, 24389, ?, 117649, 205379
Solution:
[latex]9^{3}[/latex], [latex]19^{3}[/latex], [latex]29^{3}[/latex], [latex]39^{3}[/latex], [latex]49^{3}[/latex], [latex]59^{3}[/latex]

Example 3: 1000, 8000, 27000, 64000, ?
Solution:
[latex]10^{3}[/latex], [latex]20^{3}[/latex], [latex]30^{3}[/latex], [latex]40^{3}[/latex], [latex]50^{3}[/latex]

Ration Series: This type of series are based on ration series, where sequence are in form of ratio in difference between the numbers. All numbers are arranged in ratio sequence order.
Example 1: 11, 22, 44, 88, ?
Solution:
Separate each number and
= 11 x 2 = 22,
= 22 x 2 = 44,
= 44 x 2 = 88,
= 88 x 2 = 176.

Example 2: 13, 26, 412, ?, 1648
Solution:
Separate each number and
= (1+1 = 2, 3+3 = 6) = 26,
= (2+2=4, 6+6 = 12) = 412,
= (41+41=82, 2+2=4 = 824,
= (82+82=164,4+4= 8)=1648.
So, the missing term is 824.

Geometric Series: This type of series are based on ascending or descending sequence of numbers and each continuous number is obtain by multiplying or dividing the preceding number with static number.
Example 1: 3, ?, 45, 144, 585
Solution:
3 x 0 +9 = 9, 9 x 1 + 9 = 18, 18 x 2 + 9 = 45, 45 x 3 + 9 = 144, 144 x 4 + 9 = 585.

Example 2: 5, 45, 405, 3645, ?
Solution:
5 x 9 = 45, 45 x 9 = 405, 405 x 9 = 3645, 3645 x 9 = 32805.

Example 3: 73205, 6655, 605, 55, ?
Solution:
5 x 11 = 55, 55 x 11 = 605, 605 x 11 = 6655, 6655 x 11 = 73205.

Two stage Type Series: A two stage Arithmetic series is one in which the formation of arithmetic series are obtain from differences of continuous numbers themselves.
Example: 1, 3, 6, 10, 15.....
Solution:
3 - 1 = 2, 6 - 3 = 3, 10 - 6 = 4, 15 - 10 = 5....
Now, we get an arithmetic sequence 2, 3, 4, 5

Mixed Series: This type of series are more than one different order are given in a series which arranged in alternatively in a single series or created according to any non-conventional rule. This mixed series Examples are describes in separately.
Example: 11, 24, 50, 102, 206, ?
Solution:
11 x 2 = 22 +2 = 24,
24 x 2 = 48 + 2 = 50,
50 x 2 = 100 + 2 = 102,
102 x 2 = 204 + 2 = 206,
206 x 2 = 412 + 2 = 414.
So the missing number is 414.

### Samples

1. Find out the series 291, 304, 317, ?, 343
Solution:
Given series 291, 304, 317, ?, 343
First, find the common difference between the numbers
i.e. 304 - 291 = 13
317 - 304 = 13
Now, just add 13 to 317
⇒ 317 + 13 = 330
Therefore series is 291, 304, 317, 330, 343

2. Find out the series 9, 18, 36, 72, ?
Solution:
Given series is 9, 18, 36, 72, ?
Notice that every number is a multiple of 2 i.e.
9 × 2 = 18
18 × 2 = 36
36 × 2 = 72
72 × 2 = 144
Therefore, the series is 9, 18, 36, 72, 144

3. Find out the series 30, 45, 75, 120, 180, 255, ?
Solution:
Given series 30, 45, 75, 120, 180, 255, ?
First, find out the common difference between the numbers i.e.
45 - 30 = 15
75 - 45 = 30
120 - 75 = 45
180 - 120 = 60
255 - 180 = 75
Now see the pattern of common difference i.e.
15 × 1 = 15
15 × 2 = 30
15 × 3 = 45
15 × 4 = 60
15 × 5 = 75
15 × 6 + 255 = 345
Therefore, the series is 30, 45, 75, 120, 180, 255, 345

4. Find out the series 9, 90, 154, 203, 239, 264, ?
Solution:
Given series 9, 90, 154, 203, 239, 264, ?
Multiply second last number by 2 i.e. 2 × 239=478 > 264
So, take common difference between them
90 - 9 = 81
154 - 90 = 64
203 - 154 = 49
239 - 203 = 36
264 - 239 = 25
By observing the above pattern, they are square numbers i.e [latex]9^2, 8^2, 7^2, 6^2, 5^2, 4^2[/latex] respectively
So, [latex]4^2[/latex] + 264 = 280
Therefore, series is 9, 90, 154, 203, 239, 264, 280

5. Find out the series 7, 13, 25, 49, 97, 194, 385?
Solution:
Given series 7, 13, 25, 49, 97, 194, 385?
7 × 2 - 1 = 13
So, 13 × 2 - 1 = 25
25 × 2 - 1 = 49
49 × 2 - 1 = 97
97 × 2 - 1 = 193
193 × 2 - 1 = 385
Therefore, the wrong number is 194