# For our studentsLogical Reasoning

## LawpreneurzLogical Reasoning

### Introduction

ANALYSIS OF NUMERICAL SERIES: A close analysis of the above examples show that the number series can be of the following types.

• Pure Series: In this type the number follow a pattern which can be easily understood. The number itself may be:
1. Perfect square
2. Perfect cube
3. Prime number
4. Combination

Difference Series: The difference series in of the first number with the next number makes a series.

Ex: i) 1, 3, 5, 7, 9, 11, ……. The difference between two consecutive numbers is 2. So it makes a series.

Ex: ii) 1, 2, 4, 7, 11, 16……. The difference between the consecutive number is 1, 2, 3, 4, 5 and so on it also makes a perfect series.

#### COMPLETING THE GIVEN SERIES

Ex.1. Which is the number that comes next in the sequence?
1, 5, 13, 29, __ , 125

1. 32
2. 62
3. 61
4. 31

Sol. In this case, the series is increasing by +4, +8, +16, +32, +64. So our answer is 61, as by adding the number 32 to 29 we get the required number i.e. 61.

Ex.2. 1, 4, 9, 16, 25, ?

1. 35
2. 36
3. 48
4. 49

Sol.: The number are 12, 22, 33, 42, 52,
Hence 62 = 36

Ex.3. Which is the number that comes next in the sequence?
6, 11, 21, 36, 56, __

1. 42
2. 51
3. 81
4. 91

The answer here is (3) 81 because the series is progressing by factor of 5, 10, 15, 20, 25.

Ex. 4. Which is the number that comes next in the sequence?
0, 6, 24, 60, 120, 210, ?

1. 240
2. 290
3. 336
4. 504

Sol. Clearly, the given series is : 13 - 1, 23 - 2, 33 - 3, 43 - 4, 53 - 5, 63 - 6.

Next number = 7 3 – 7 = 343 – 7 = 336
Hence the answer is (3)

Ex. 5. Which is the number that comes next in the sequence?
3, 7, 15, 31, 63, ?

1. 92
2. 115
3. 127
4. 131

Sol. Each number in the series is the preceding number multiplied by 2 and then increased by 1. Thus, (3 x 2) + 1 = 7, (7 x 2) + 1 = 15, (15 x 2) + 1 = 31 and so on
Missing number = (63 x 2) + 1 = 127
Hence the answer is (3)

Ex. 6. 3, 6, 18, 72, ?

1. 144
2. 216
3. 288
4. 360

Sol. The pattern is x 2, x 3, x 4, ____
Missing number = 72 x 5 = 360, i.e. answer is 360.

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 particular sequence of numbers or alphabets and on the basis of that 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.

#### Examples

1: Where the difference between two consecutive terms involves various arithmetic operation.

1. 2, 5, 8, 11, ……
The common difference is +3.
2. 33, 25, 17, 9, ……
The common difference is -8.
3. 4, 8, 24, ……
Here, the series in multiple of 4.
4. 52, 26, 13, ……
The series is based on the rule that the every number is divided by 2 to get the next number.

2: Where the difference between the consecutive numbers is in some progression, i.e. A.P. or G.P.

1. 7, 18, 34, 55, ……
Here, the difference between the terms is +11, +16, +21, which are in A.P.
2. 8, 9, 12, 21, 48, 129, ……
Here, the difference between the terms is +1, +3, +9, +24, +81. Here, these terms are in G.P.

3: Where series terms differ each other by perfect squares and cubes.

1. 5, 6, 10, 19, 35, ……
The common difference between the number of series is +(1)2, +(2)2, +(3)2, +(4)2 ……
2. 7, 8, 16, 43, 107, ……
The common difference between the term of series is 107-43, 43-16, 16-8, 8-7
(4)3, (3)3, (2)3, (1)3.
3. 1, 4, 9, 16, ……
The term of series are (1)2, (2)2, (3)2, (4)2, ……
4. 27, 64, 125, 216, ……
The term of series are (3)3, (4)3, (5)3, (6)3, ……

### Exercise

1. 5, 14, 27, 44, 65, ?
2. 3, 4, 10, 33, 136, ?
3. 5, 6, 9, 18, 45, ?
4. 7, 24, 75, 228, ?
5. 17, 68, 612, ?
6. 84, 64, 46, 30,?
7. 2, 2, 4, 4, 6, 8, 8, ?
8. 3, 9, 9, 36, 180, ?
9. 1, 8, 9, 64, 25, 216, ?, ?
10. 4, 13, 53, 266, 1597, ?
11. 81, 72, 63, ?, 45
12. 7, 13, 21, ?, 43, 57
13. 3, 15, 35, ?, 99, 143
14. 9, 11, 15, 23, ?
15. 8, 15, 28, 53, ?

### Solutions

1. The difference in the successive terms of the series is increasing by 4. That is, 5 + 9 = 14, then 14 + 13 = 27, then 27 + 17 = 44, 65 + 25 = 90.
2. From the first term onwards, the series is x 1 +1, x 2 +2, x 3 +3, x 4 +4 and finally it is 136 x 5 + 5 = 685.
3. The difference between the consecutive terms goes on becoming 3 times.
Thus, 45 + 27 x 3 = 45 + 18 = 126.
4. The series is 7 +17 = 24. Then, 24 + 51 (17x3) = 75, 75 + 153 (51x3) = 228.
So, now it is 228 + (153 x 3) = 687. OR 7 x 3 +3 = 24, 24 x 3 + 3 = 75, 75 x 3 + 3 = 228. Hence next is 228 x 3 +3 = 687.
5. The series is 17 x 12 = 17, 17 x 22 = 68, 68 x 32 = 612. Hence 612 x 42 = 9792.
6. The pattern of the series is 84 – 20 = 64, 64 – 18 = 46, 46 -16 = 30 and hence, now it is 30 – 14 = 16.
7. This is an alternative series arrangement. The odd terms is an even number series. The even terms is a double series, that is 2 x 2 = 4, 4 x 2 = 8, 8 x 2 = 16.
8. The pattern of the series is 3 x 3 = 9, 9 x 4 = 36, 36 x 5= 180 and finally 180 x 6 = 1080.
9. This is an alternative series arrangement. The odd terms is a series of square 3, 5, 7. The even terms is a series of cubes of numbers 2, 4, 6 and 8. Hence, the answer is 49, 512.
10. The series is x 3 + 1, x 4 + 1, X 5 + 1 and so on.
11. All the terms differ by the number 9. So, the answer is 54.
12. The difference in the terms of the series is starting from the first term is +6, +8, +10, +2 and +14.
13. The pattern of the series is 3 + 12 = 15, 15 + 20 = 35, so now it is 35 + 28 = 63, then 63 + 36 = 99. Hence the difference itself differs by +8.
14. The pattern of the series is 9 + 2 = 11, 11 + 4 = 15, 15 + 8 = 23 and finally, it is 23 + 16 = 39.

15. 13 = 7 x 2 - 1, 25 = 13 x 2 - 1
Hence, the next term is 49 = 25 x 2 - 1.

### Exercise

Question : In this questions, a number series is given with one term missing. Choose the correct alternative that will continue the same pattern and fill in the black spaces.

Q. 1, 4, 9, 16, 25, (____)

1. 5
2. 36
3. 48
4. 49

Question : In this questions, a number series is given with one term missing. Choose the correct alternative that will continue the same pattern and fill in the black spaces.

Q. 20, 19, 17, (____), 10, 5

1. 12
2. 13
3. 14
4. 15

Question : In this questions, a number series is given with one term missing. Choose the correct alternative that will continue the same pattern and fill in the black spaces.

Q. 2, 3, 5, 7, 11, (_), 17

1. 12
2. 13
3. 14
4. 15

Question : In this questions, a number series is given with one term missing. Choose the correct alternative that will continue the same pattern and fill in the black spaces.

Q. 6, 11, 21, 36, 56, (___)

1. 42
2. 51
3. 81
4. 91

Question : In this questions, a number series is given with one term missing. Choose the correct alternative that will continue the same pattern and fill in the black spaces.

Q. 3, 9, 27, 81, (___)

1. 324
2. 210
3. 243
4. 162

Question : In this questions, a number series is given with one term missing. Choose the correct alternative that will continue the same pattern and fill in the black spaces.

Q. 2, 5, 9,(___), 20, 27

1. 16
2. 14
3. 18
4. 24

Question : In this questions, a number series is given with one term missing. Choose the correct alternative that will continue the same pattern and fill in the black spaces.

Q. 5, 9, 17, 29, 45, (___)

1. 65
2. 60
3. 70
4. 68

Question : In this questions, a number series is given with one term missing. Choose the correct alternative that will continue the same pattern and fill in the black spaces.

Q. 1, 6, 15, (___), 45, 66, 91

1. 25
2. 26
3. 27
4. 28

Question : In this questions, a number series is given with one term missing. Choose the correct alternative that will continue the same pattern and fill in the black spaces.

Q. 1, 2, 3, 5, 8, (___)

1. 13
2. 11
3. 9
4. 15

Question : In this questions, a number series is given with one term missing. Choose the correct alternative that will continue the same pattern and fill in the black spaces.

Q. 0.5, 1.5, 4.5, 13.5, (___)

1. 30.5
2. 45.5
3. 39.5
4. 40.5

Question : In this questions, a number series is given with one term missing. Choose the correct alternative that will continue the same pattern and fill in the black spaces.

Q. 121, 225, 361,(___)

1. 529
2. 441
3. 484
4. 729

Question : In this questions, a number series is given with one term missing. Choose the correct alternative that will continue the same pattern and fill in the black spaces.

Q. 19, 2, 38, 3, 114, 4, (___)

1. 228
2. 456
3. 256
4. 352

Question : In this questions, a number series is given with one term missing. Choose the correct alternative that will continue the same pattern and fill in the black spaces.

Q. 1, 2, 3, 6, 9, 18, (___), 54

1. 27
2. 18
3. 36
4. 81

Question : In this questions, a number series is given with one term missing. Choose the correct alternative that will continue the same pattern and fill in the black spaces.

Q. 12, 32, 72, 152, (___)

1. 312
2. 325
3. 515
4. 613

Question : In this questions, a number series is given with one term missing. Choose the correct alternative that will continue the same pattern and fill in the black spaces.

Q. 3, 6, 5, 20, 7, 42, 9, (___)

1. 72
2. 66
3. 60
4. 54

Question : In this questions, a number series is given with one term missing. Choose the correct alternative that will continue the same pattern and fill in the black spaces.

Q. 1, 3, 4, 8, 15, 27, (___)

1. 37
2. 44
3. 50
4. 55

Question : In this questions, a number series is given with one term missing. Choose the correct alternative that will continue the same pattern and fill in the black spaces.

Q. 95,115.5, 138, (___), 189

1. 162.5
2. 154.5
3. 164.5
4. 166.5

Question : In this questions, a number series is given with one term missing. Choose the correct alternative that will continue the same pattern and fill in the black spaces.

Q. 24, 60, 120, 210, (___)

1. 336
2. 330
3. 420
4. 525

Question : In this questions, a number series is given with one term missing. Choose the correct alternative that will continue the same pattern and fill in the black spaces.

Q. 5, 17, 37, 65, (____), 145

1. 97
2. 95
3. 99
4. 101

Question : In this questions, a number series is given with one term missing. Choose the correct alternative that will continue the same pattern and fill in the black spaces.

Q. 34, 18, 10, 6, 4, (___)

1. 3
2. 2
3. 1
4. 0

Question : In this questions, a number series is given with one term missing. Choose the correct alternative that will continue the same pattern and fill in the black spaces.

Q. 1, 2, 5, 12, 27, 58, 121, (___)

1. 246
2. 247
3. 248
4. 249

Question : In this questions, a number series is given with one term missing. Choose the correct alternative that will continue the same pattern and fill in the black spaces.

Q. 0.5, 0.55, 0.65, 0.8, (___)

1. 0.9
2. 0.82
3. 1
4. 0.95

Question : In this questions, a number series is given with one term missing. Choose the correct alternative that will continue the same pattern and fill in the black spaces.

Q. 17, 19, 23, 29, (___), 37

1. 36
2. 35
3. 33
4. 31

Question : In this questions, a number series is given with one term missing. Choose the correct alternative that will continue the same pattern and fill in the black spaces.

Q. 3, 12, 27, 48, 75, 108, (___)

1. 192
2. 183
3. 147
4. 162

Question : In this questions, a number series is given with one term missing. Choose the correct alternative that will continue the same pattern and fill in the black spaces.

Q. 563, 647, 479, 815, (___)

1. 672
2. 386
3. 143
4. 279

1. B
2. C
3. B
4. C
5. C
6. B
7. A
8. D
9. A
10. D
11. A
12. B
13. A
14. A
15. A
16. C
17. A
18. A
19. D
20. C
21. C
22. D
23. C
24. C

## Lawpreneurz Details of lecture

Sr No. Topics
Lecture 1 Alphabetical Series
Lecture 2 Numerical Series
Lecture 3 Coding Decoding
Lecture 4 Directions
Lecture 5 Blood Relations
Lecture 6 Analytical Reasoning ( Tabular Arrangement, Seating Arrangement)
Lecture 7 Coded Inequality
Lecture 8 Facts Inferences and Judgements
Lecture 9 Arguments
Lecture 10 Statement and Assumptions
Lecture 11 Statement and Conclusions
Lecture 12 Syllogism
Lecture 13 Logical Consistency
Lecture 14 Assertion and Reason
Lecture 15 Course of Action
Lecture 16 Clock
Lecture 17 Calendar
Lecture 18 Logical Connectives

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