1. Number Series
1.1 Introduction
1.2 Differences Series
1.3 Product series
1.4. Squares/Cubes series
1.5 Combination series
1.6 Miscellanous series
1.1 Introduction
For better understading, we classify this into the following categories.
1. Difference series
2. Product series
3. Squares / Cubes series
4. Combination series
5. Miscellaneous series
1.2 Difference series
The differece series can be further, classified as follows:
(a) Number series with a constant difference ; Here the difference between two consecutive numbers is always constant. For example, the numbers of the series 1, 4, 7, 10, 13 ..... are such that the difference between two consecutive terms is constant. Here this difference is 3.
(b) Number series with increasing /decreasing difference : Here the difference between consecutive terms is not constant. It either decreases or increases, e.g. the series 10, 15, 19, 22, 24 ....
Here the difference between 1st & 2nd terms, 2nd & 3rd terms, 3rd & 4th terms and so on are 5, 4, 3, 2 respectively.
Since the difference between 22 & 24 is 2, the next difference should be 1. So, the number that comes after 24 should be 25.
1.3 Product series
Consider the series 3, 6, 12, 24, 48, 96.
Here, each number in the series is multiplied by 2 to get the next term. So, the term that comes after 96 is 192.
Similarly we can have a series where numbers are obtained by dividing the previous term with a constant number, e.g. consider the series 81, 27, 9, 3..... Here, each term is obtained by dividing the previous term by 3 (or in other words by multiplying the previous term by 1/3).
Therefore next term will be 3 × = 1
Consider the series 5, 25, 100, 300.......
Here, the first term is multiplied by 5 to get the second term. The second term is multiplied by 4 to get the third term. The third term is multiplied by 3 to get the fourth term. Therefore to get the fifth term, we have to multiply fourth term by 2, i.e.the fifth term is 600. Here each term is multiplied by decreasing factor (or it could also be an increasing factor) to get the next term.
1.4 Squares/Cubes series
There can be a series where all the terms are related to the square of numbers or cube of numbers. There can be many variations in such series. For example, each term of the series may be the square of a natural number such as 1, 4, 9, 16 ....
Here the term that follows 16 will be square of 5, i.e. 25.
The terms of the series may be the square of odd number, e.g. 1, 9, 25, 49 ..... or even numbers e.g. 4, 16, 36, 64......
1.5 Combination series
This is a type of series where more than one arithmetic operation is performed.
Let us take an example. 1, 3, 7, 2, 6, 10, 3, 9, 13 ......
Here the first term is multiplied by 3 to get the second term. To get the third term we add 4 to the second term. To get the 4th term, we add 1 to the first term. After this the cycle will be repeated.
This terms 5th term = 2 × 3 = 6
6th term = 6 + 4 = 10
7th term = 2 + 1 = 3 and so on.
Consider another series, 1, 2, 6, 21, 88 ....
Here, we can observe that 88 is close to the 4 times of 21 or it is 21 × 4 + 4. Similarly 21 is 6 × 3 + 3, 6 is 2 × 2 + 2 and 2 is 1 × 1 + 1.
1.6 Miscellaneous series
Take the series 1, 3, 5, 7, 9, 11, 13 ...... It is a series of odd numbers. So the next term will be 15. There can be many variations in miscellaneous series e.g. 2, 12, 30, 56, 90, 132 .....
This is series of product of two series, as 1 × 2, 3 × 4, 5 × 6, 7 × 8, 9 × 10, 11 × 12. We can explain this series as product of odd number and even number series.
Other form of number series
Here a series of numbers is given. These numbers need not (and very often, do not) follow any specific pattern. The objective here is not to find out a missing term, but to find out how many times a given condition is satisfied in the given series.
Illustration 1
In the following number sequence how many 4’s are there that are immediately preceded by 6 and immediately followed by 5 : 342654364564538746821764586459745
Solution
In the given number sequence
3 4 2 6 5 4 3 3 8 7 4 6 8 2 1 7 8 9 7 4 5
Solved Examples
Directions: The terms given in each of the following questions follow a definite pattern and thus make a series.
Find the missing number from the series out of the given option (A) to (D):
Example 1
3, 6, 18, 72, (......)
(A) 144 (B) 216 (C) 288 (D) 360
Solution
(d) The pattern is × 2, × 3, × 4,
Missing number = 72 × 5 = 360.
Example 2
19, 2, 38, 3, 114, 4, (......)
(A) 228 (B) 256 (C) 352 (D) 456
Solution
(d) The sequence is a combination of two series :
I. 19, 38, 114 (......) and 1, 2, 3, 4
The pattern followed in I is × 2, × 3,
Missing number = 114 × 4 = 456.
Example 3
121, 225, 361, (......)
(A) 441 (B) 484 (C) 529 (D) 729
Solution
(c) The numbers are 112, 152, 192, i.e.112,
(11+4×1)2 (11+4×2)2,
Missing number = (11 + 4 × 3)2 = (23)2 = 529.
Example 4
0.5, 1.5, 4.5, 13.5, (......)
(A) 9 (B) 11 (C) 13 (D) 40.5
(d) Each term of the series is obtained by multiplying the preceding term by 3.
Missing number = 13.5 × 3 = 40.5.
Example 5
1, 2, 3, 5, 8, (......)
(A) 9 (B) 11 (C) 13 (D) 15
Solution
(c) Each term in the series is the sum of the preceding two terms.
Example 6
1, 6, 15, (......) 45, 66, 91
(A) 25 (B) 26 (C) 27 (D) 28
Solution
(d) The pattern is +5, +9, , +21, +25.
Missing number = 15 + 13 = 28.
Example 7
5,9, 17, 29, 45, (......)
(A) 60 (B) 65 (C) 68 (D) 70
Solution
(b) The pattern is +4, +8, +12, +16, .,
Missing number = 45+20 = 65.
Example 8
2, 5, 9, (......), 20, 27
(A) 14 (B) 16 (C) 18 (D) 24
Solution
(a) The pattern is +3, +4, Missing number = 9+5 = 14.
Example 9
1, 9, 17, 33, 49, 73, (......)
(A) 97 (B) 98 (C) 99 (D) 100
Solution
(a) Thepatternis +8, +8 + 16, +16, +24, ....
Missing number = 73+24 = 97.
Example 10
3, 9, 27, 81, (......)
(A) 324 (B) 243 (C) 210 (D) 162
Solution
(b) Each term of the given series is obtained by multiplying its preceding term by 3.
Missing number = 81 x 3 = 243
Example 11
1, 6, 13, 22, 33 (......)
(A) 44 (B) 45 (C) 46 (D) 47
Solution
(c) The pattern is +5, +7, +9, +11
Missing number = 33 + 13 = 46.
Example 12
6, 11, 21, 36, 56, (......)
(A) 42 (B) 51 (C) 81 (D) 91
Solution
(c) The pattern is +5, +10, +15+20,
Missing number = 56+25 = 81.
Example 13
20, 19, 17, (......), 10, 5
(A) 12 (B) 13 (C) 14 (D) 15
Solution
(c) The pattern is -1, -2,
Missing number = 17 - 3 = 14.
Example 14
1, 4, 9, 16, 25, (......)
(A) 35 (B) 36 (C) 48 (D) 49
Solution
(b) The numbers are I2, 22, 32, 42, 52.
Missing number = 62= 36.
Example 15
462, 420, 380, (......), 306
(A) 322 (B) 332 (C) 342 (D) 352
Solution
(c) The pattern is –42, –40,
Missing number =380 – 38 = 342.
Example 16
5, 16, 49, 104, (......), 280
(A) 115 (B) 148 (C) 170 (D) 181
Solution
(d) The pattern is + 11, + 33, + 55, ........, i.e.+ (11 × 1), + (11 × 3), + (11 × 5),
Missing number = 104 + (11 × 7) = 181.
Example 17
9, 11, 20, 31, (......), 82
(A) 41 (B) 51 (C) 60 (D) 71
Solution
(b) Each term in the series is the sum of the preceding two terms.
Missing number = 20 + 31 = 51.
Example 18
5, 17, 37, 65, (......), 145
(A) 95 (B) 97 (C) 99 (D) 101
Solution
(d) The numbers are 22 + 1, 42 + 1, 62 + 1, 82 + 1,.........122 + 1.
Missing number = 102 + 1 =101.
Example 19
24, 60, 120, 210, (......)
(A) 300 (B) 336 (C) 420 (D) 525
Solution
(b) The pattern is + 36, + 60, + 90...............i.e. + [6 × (6 + 0)], [6 × (6 + 4)], [6× (6 + 9)],....
Example 20
4, 10, (......), 82, 244, 730
(A) 24 (B) 28 (C) 77 (D) 218
Solution
(b) Each number in the series is the preceding number multiplied by 3 and then decreased by 2.
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EXERCISE-I
Directions : Find the number at the empty space.
1. 19, 24, 20, 25, 21, 26, ? 2. 11, 14, 12, 15, 13, 16, ?
3. 10, 2, 8, 2, 6, 2, ? 4. 8, 9, 1 l, 14, 18, 23, ?
5. 25, 25, 22, 22, 19, 19, ? 6. 14, 2, 12, 4, 10, 6, ?
7. 7, 16, 9, 15, 11, 14, ? 8. 40, 42, 39, 44, 38, 46, ?
9. 3, 18, 4, 24, 5, 30, ? 10. 18, 20, 22, 20, 28, 20 , ?
11. 18, 20, 10, 12, 4, 6, ? 12. 7, 6, 8, 5, 3, 7, ?
13. 17, 19, 23, 29, ?, 37 14. 563, 647, 479, 815, ?
15. 2, 9, 28, 65, ? 16. 4, 8, 16, 32, 64, 128, ?
17. 8, 16, 24, 32, 40, 48, ? 18. 13, 11, 14, 12, 15, 13, ?
19. 6, 18, 36, 108, 216, 648, ? 20. 4, 4, 8, 8, 16, 16, ?
21. 2, 6, 18, 54, 162, 486, ? 22. 4, 20, 35, 49, 62, 74, ?
23. 10, 18, 15, 23, 20, 28, ? 24. 4, 10, 8, 14, 12, 18, ?
25. 10, 15, 12, 17, 14, 19, ?
EXERCISE-II
1. 14, 17, 20,_____,26, 29
(A) 21 (B) 25 (C) 23 (D) 24
2. 5, 11, 17, 23,————35
(A) 27 (B) 29 (C) 25 (D) 31
3. 48, 43, 39,______, 34, 33
(A) 35 (B) 37 (C) 38 (D) 36
4. 6, 8, 12, 18, 26,_____
(A) 36 (B) 38 (C) 34 (D) 32
5. 63, 66, 71, 78, 87,_____
(A) 108 (B) 98 (C) 88 (D) 96
6. 43, 44, 48, 57, 73,_____
(A) 92 (B) 104 (C) 84 (D) 98
7. 4, 49, 144, 289_____
(A) 529 (B) 441 (C) 484 (D) 361
8. 4, 8, 12, 7, 11, 18, 9, _____, 22
(A) 11 (B) 15 (C) 13 (D) 7
9. 2, 4, 8, 3, 9, 27, 5, 25, 125_____, ____, ____.
(A) 6, 36, 216 (B) 8, 64, 512 (C) 9, 81, 100 (D) 7, 49, 343
10. 11,12,13, 10, 15, 8,_____,6.
(A) 17 (B) 6 (C) 16 (D) 19
11. 4,16, 64, 256,_____,4096
(A) 1024 (B) 1000 (C) 1248 (D) 894
12. 7, 21, 22, 66, 67,_____, 202
(A) 68 (B) 201 (C) 176 (D) 198
13. 49,1625, 3649, 6481,_____
(A) 81100 (B) 100144 (C) 100121 (D) 121169
14. , , , , _____.
(A) (B) (C) (D)
15. 3, 3, 6, 18, 72, 360, _____.
(A) 2160 (B) 2430 (C) 1880 (D) 2040
16. 8, 40, 20, 100, 50, _____, 125.
(A) 225 (B) 250 (C) 200 (D) 300
17. 43, 47, 53, 59, 61, _____.
(A) 63 (B) 65 (C) 69 (D) 67
18. 67, 71, 73, 79, 83,_____,_____
(A) 87,89 (B) 89,91 (C) 89,97 (D) 93,97
19. 47, 51, 57, 65,_____87
(A) 83 (B) 81 (C) 79 (D) 75
20. 113,136,161,188,____,248
(A) 213 (B) 217 (C) 223 (D) 219
EXERCISE-III
1. 43, 47, 90, 56, 63, 119, 67, 79,______.
(A) 146 (B) 147 (C) 143 (D) 153
2. 13, 221, 17, 19, 437, 23, 23,_____29
(A) 617 (B) 667 (C) 697 (D) 707
3. 24, 30, 36, 42, 52, 60______
(A) 76 (B) 64 (C) 90 (D) 68
4. 343, 1331, 2197, 4913, 6859______
(A) 12167 (B) 9261 (C) 10648 (D) 11739
5. 64, 216, 512, 1000,1728,_____
(A) 2197 (B) 3375 (C) 3125 (D) 2744
6. 2, 8, 4, 64, 7, 343, 11, 1331, 16______
(A) 4286 (B) 3916 (C) 4096 (D) 4196
7. 1, 8 ,9, 64, 25, 216, 49,_____
(A) 64 (B) 729 (C) 512 (D) 81
8. 1728, 1385, 1169, 1044, 980,_____
(A) 972 (B) 916 (C) 954 (D) 953
9. 17, 34, 102, 408, 2040,_____
(A) 13220 (B)12240 (C) 12420 (D) 12680
10. 12,14, 18, 26, 38, 62______
(A) 76 (B) 72 (C) 74 (D) 80
11. 12, 36, 150, 392, 1452,_____
(A) 2452 (B) 2197 (C) 2246 (D) 2366
12. 13, 51, 203, 811, 3243,_____
(A) 19972 (B)12971 (C) 12876 (D) 12761
13. 1584, 900______,180, 48, 4
(A) 448 (B) 576 (C) 504 (D) 478
14. 28, 38, 49______70, 77
(A) 58 (B) 64 (C) 66 (D) 62
15. 49, 925, 2549,_____121169
(A) 64121 (B) 81121 (C) 100121 (D) 49121
16. 11, 22, 88______, 18304, 292864
(A) 1408 (B) 1628 (C) 1418 (D) 1348
17. 3, 35, 99,195,_____,483
(A) 343 (B) 323 (C) 353 (D) 363
18. 13,14, 22, 49, 113______454
(A) 248 (B) 224 (C) 256 (D) 238
19. 21, 23, 46, 48, 96, 98,_____
(A) 100 (B) 186 (C) 196 (D) 174
20. 900, 18, 180, 36, 45, 108,_____, 432
(A) 25 (B) 15 (C) 9 (D) 7
21. 930, 812, 702, 600______
(A) 506 (B) 380 (C) 342 (D) 294
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Answers
Exercise-I
1. 22 2. 14 3. 4 4. 29 5. 16
6. 8 7. 13 8. 37 9. 6 10. 36
11. 0 12. 4 13. 31 14. 143 15. 126
16. 256 17. 56 18. 16 19. 1296 20. 32
21. 1458 22. 85 23. 25 24. 16 25. 16
Exercise-II
1. (C) 2. (B) 3. (D) 4. (A) 5. (B)
6. (D) 7. (C) 8. (C) 9. (D) 10. (A)
11. (A) 12. (B) 13. (C) 14. (B) 15. (A)
16. (B) 17. (D) 18. (C) 19. (D) 20. (B)
Exercise-III
1. (A) 2. (B) 3. (D) 4. (A) 5. (D)
6. (C) 7. (C) 8. (D) 9. (B) 10. (C)
11. (D) 12. (B) 13. (A) 14. (D) 15. (D)
16. (A) 17. (B) 18. (D) 19. (C) 20. (B)
21. (A)
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