Bank probationary Officer
Quantitative Aptitude
SERIES
A series is a sequence of numbers i. Two - tier Arithmetic series - A two
where the sequence of numbers is obtained by tier Arithmetic series shall be one in which the
some particular predefined rule and applying that differences of successive numbers themselves
predefined rule it is possible to find out the next form an arithmetic series.
term of the series. Eg: i. 1, 2, 5, 10, 17, 26, 37, ----
A series can be created in many ways. ii. 3, 5, 9, 15, 23, 33, ----
Some of these are as explained below:
Two - tier arithmetic series can be denoted as a
1. Arithmetic series:- An arithmetic se- quadratic function, which can be denoted as
ries is one in which successive numbers are f ( x) x 2 1, where x 0, 1, 2,
obtained by adding or subtracting a fixed number ii. Three - tier Arithmetic series:
to the previous number
This as the name suggest is a series in
Eg: i. 5, 9, 13, 17, -------- which the differences of successive numbers
ii. 35, 31, 27, 23, 19, ------ form a two - tier arithmetic series, whose suc-
2. Geometric Series: A geometrical se- cessive terms differences, intern, form an arith-
ries is one in which each successive number is metic series.
obtained by multiplying or dividing a fixed number Eg: (i) 336, 210, 120, 60, 24, 6, 0, ---
by the previous number. Here the differences in successive terms
Eg: i. 4, 8, 16, 32, 64, ------ are 126, 90, 60, 36, 18, 6, ----
ii. 15, -30, 60, -120, ----- The differences of successive terms of
iii. 3125, 625, 125, 25, 5, ----- this new series are 36, 30, 24, 18, 12, ----- which
3. Series of squares, cubes etc:- is an arithmetic series.
These series can be formed by squar- The three-tier arithmetic series can be
ing or cubing every successive number. denoted as a cubic function, which can be de-
noted as f ( x) x 3 x, where x 1, 2,
Eg: i. 2, 4, 16, 256, -----
(iii) We know that,
ii. 2, 8, 512, ------
(a) In an arithmetic series, we add or
4. Mixed Series: deduct a fixed number to find the next number,
A mixed series is basically the one we and
need to have sound practice of because it is (b) In a geometric series we multiply or
generally the mixed series which is asked in divide a fixed number to find the next number.
the examinations. By a mixed series, we mean
We can combine these two ideas into
a series which is created according to any non-
one to form.
conventional (but logic) rule. Because there is
no limitations to people imagination, there are 1. Arithmetico - Geometric Series: As
infinite ways in which a series can be created and the name suggests in this series each succes-
naturally it is not possible to club together all of sive term should be found by first adding a fixed
them. Still we are giving examples of some more number to the previous term and then multiply-
popular ways of creating these mixed series. ing it by another fixed number.
Eg: 1, 6, 21, 66, 201, -------- Step II. Check Trend : Increasing/Decreasing
Here each successive term is obtained Alternating
by first adding 1 to the previous term and then If you fail to see the rule of the series
multiplying it by 3. The differences of succes- by just preliminary screening you should see
sive numbers should be in Geometric progres- the trend of the series. By this we mean that
sion. you should check whether the series increases
2. Geometrico-Arithmetic Series :- As continuously or decreases continuously or
the name suggests, a geometrico - Arithmetic whether it alternates ie., increases and de-
series should be the one in which each succes- creases alternating.
sive term is found by first multiplying or dividing Step III : Use the following rules:
the previous term by a fixed number and then 1. If the rise of series is slow or gradual,
adding or deducting another fixed number. the series is likely to have an addition - based
Eg: 3, 4, 7, 16, 43, 124 ------ increase, successive numbers are obtained by
Here each successive term is obtained adding some numbers.
by first multiplying the previous number by 3 and 2. If the rise of a series is very sharp ini-
then subtracting 5 from it. tially but slows down later on, the sereis is likely
The differences of successive numbers to be formed by adding squared or cubed num-
should be in geometric progression. bers
(iv) Twin Series: As the name suggests, 3. If the rise of a series is throughout
a twin series are two series packed in one. equally sharp, the series is likely to be multipli-
cation-based; successive terms are obtained by
Eg: 1, 3, 5, 1, 9, -3, 13, -11, 17, -----
multiplying by some terms (and, maybe, some
Here the first, third, fifth, etc. terms are addition or subtraction could be there, too)
1, 5, 9, 13, 17, which is an arithmetic series.
4. If the rise of a series is irregular and
The second, fourth, sixth, etc. are 3, 1, -3, -11
haphazard there may be two possibilities. Ei-
which is a geometrico -arithmetic series in which
ther there may be a mix of two series or two
successive terms are obtained by multiplying
different kinds of operations may be going on
the previous term by 2 and then subtracting 5.
alternatively. The first is more likely when the
(v) Other series: Besides, numerous other increase is very irregular, the second is more
series are possible and it is impossible to even likely when there is a pattern, even in the irregu-
think of all of them. It is only through a lot of larity of the series.
practice and by keeping abreast with the latest
Finding wrong Numbers is a series
trends that one can expect to master the se-
ries. In today's examinations, a series is likely to be
given in format of a complete series in which an
Steps for Solving Series Questions:-
incorrect number is included. The candidate is
Despite the fact that it is extremely dif- required to find out the wrong number.
ficult to lay down all possible combinations of
Obviously, finding the wrong number in a series
series, still, if you follow the following step-by-
is very easy once you have mastered the art of
step approach, you may solve a series ques-
understanding how the sereis is likely to be
tions easily and quickly.
formed. On studying a given series and apply-
Step 1 : Preliminary Screening: ing the concepts employed so far you should
First check the series by having a look be able to understand and thus decode the for-
at it. It may be noted that the series is very sim- mation of the series. This should not prove very
ple and just a first look may be enough and you difficult because usually six terms are given and
may know the next term. it means that at least five correct terms are given.
This should be sufficient to follow the series.
PRACTICE TEST
Direction (Qs. 1-10) : Each of the following number series contains a wrong number. Find out that
number.
1. 527 318 237 188 163 154 150
(1) 318 (2) 237 (3) 188 (4) 163 (5) 154
2. 14 40 77 229 455 1367 2723
(1) 40 (2) 77 (3) 229 (4) 455 (5) 1367
3. 20250 3375 1350 225 75 15 6
(1) 3375 (2) 1350 (3) 225 (4) 75 (5) 15
4. 80 81 83 90 100 115 136
(1) 81 (2) 83 (3) 90 (4) 100 (5) 115
5. 8 15 50 250 1100 5475 27350
(1) 15 (2) 50 (3) 250 (4) 1100 (5) 5475
6. 70.21 71.49 71.81 70.85 72.13 71.17
(1) 71.49 (2) 71.81 (3) 70.85 (4) 72.13 (5) 70.56
7. 10 21 64 255 1286 7717 54020
(1) 21 (2) 255 (3) 64 (4) 1286 (5) 7717
8. 3 4.5 9 22.5 67.5 270 945
(1) 4.5 (2) 9 (3) 22.5 (4) 67.5 (5) 270
9. 376 188 88 40 16 4 -2
(1) 188 (2) 88 (3) 40 (4) 16 (5) 4
10. 5 6 10 37 56 178
(1) 6 (2) 56 (3) 37 (4) 10 (5) 178
Directions (Q. 11- 20) in each of the following questions, a number series is established if the posi-
tions of two out of the five marked numbers are interchanged. The position of the first unmarked
number remains the same and it is the beginning of the series. The earlier of the two marked numbers
whose positions are interchanged is the answer. For example, if an interchange of the number marked
'1' and the number marked '4' is required to establish the series, your answer is '1'. If it is not neces-
sary to interchange the position of the numbers to establish the series, give 5 as your answer. Re-
member that when the series is established, the numbers change from left to right (ie from the un-
marked number to the last marked number) in a specific order.
11. 2 11 59 227 697 1369
(1) (2) (3) (4) (5)
12. 379 500 591 556 267 331
(1) (2) (3) (4) (5)
13. 0 1 -2 68 21 465
(1) (2) (3) (4) (5)
14. 11880 1225.125 10890 1089 9801 990
(1) (2) (3) (4) (5)
15. 25 26 710 175 56 3563
(1) (2) (3) (4) (5)
16. 192 24 28 168 140 35
(1) (2) (3) (4) (5)
17. -1 0 -2 -15 -6095 -236
(1) (2) (3) (4) (5)
18. 739 547 635 106 10 186
(1) (2) (3) (4) (5)
19. 0 0.25 1.50 6.75 31 161.25
(1) (2) (3) (4) (5)
20. 714 125 2 9 0 20
(1) (2) (3) (4) (5)
Directions (Qs. 21-30): In each of the following questions a number series is given. After the series, a
number is given below it, followed by A,B,C,D and E. You have to complete the series starting with the
number given and following the same property as in the given number series. Then answer the ques-
tions below it.
21. 123 149 182 224 277
321 A B C D E
Find the value of E
(1) 532 (2) 558 (3) 528 (4) 545 (5) None of these
22. 1 5 14 39 88
4 A B C D E
What should replace B?
(1) 43 (2) 17 (3) 34 (4) 40 (5) None of these
23. 2520 280 2240 320 1920 384
504 A B C D E
What should come in the place of C.
(1) 448 (2) 384 (3) 74 (4) 120 (5) 64
24. 6 9 21 39 83 163 333
4 A B C D E
Find the value of D
(1) 61 (2) 51 (3) 57 (4) 49 (5) None of these
25. 659 130 491 266 387 338
1009 A B C D E
What should come in the place of (D)
(1) 616 (2) 737 (3) 762 (4) 726 (5) None of these
26. 6 3 1 -4 -25
7 A B C D E
Find the value of C.
(1) -6 (2) -3 (3) -2 (4) 4 (5) None of these
27. 7 42 48 336 343
3 A B C D E
What is the value of D?
(1) 181 (2) 175 (3) 167 (4) 168 (5) None of these
28. 198 166 144 130 122
263 A B C D E
What should replace E?
(1) 183 (2) 178 (3) 182 (4) 180 (5) None of these
29. 8 17 44 125 368
4 A B C D E
Find the value of B.
(1) 8 (2) 42 (3) 22 (4) 9 (5) 39
30. 4 10 17 49 95
6 A B C D E
What is the value of C?
(1) 81 (2) 83 (3) 87 (4) 85 (5) None of these
Directions (Qs. 31-40): One number is wrong in each of the number series given in each of the
following questions. You have to identify that number and assuming that a new series starts with that
number following the same logic as in the given series, which of the numbers given in (1), (2), (3), (4)
and (5) given below each series will be third number in the new series?
31. 3 5 12 38 154 914 4634
(1) 1636 (2) 1222 3) 1834 (4) 3312 (5) 1488
32. 3 4 10 34 136 685 4116
(1) 22 (2) 276 (3) 72 (4) 1374 (5) 12
33. 214 18 162 62 143 90 106
(1) -34 (2) 110 (3) 10 (4) 91 (5) 38
34. 160 80 120 180 1050 4725 25987.5
(1) 60 (2) 90 (3) 3564 (4) 787.5 (5) 135
35. 572 4600 576 4032 672 3352
(1) 3371 (2) 3375 (3) 26397 (4) 4399.5 (5) None of these
36. 7 14 42 165 840 5040
(1) 330 (2) 165 (3) 990 (4) 3960 (5) None of these
37. 72 143 287 570 1147
(1) 4557 (2) 2289 (3) 1139 (4) 573 (5) None of these
38. 1 5 21 57 120 221
(1) 140 (2) 120 (3) 124 (4) 176 (5) None of these
39. 5 17 27 60 115
(1) 247 (2) 501 (3) 127 (4) 60 (5) None of these
40. 10 6 15 120 1879
(1) 116 (2) 120 (3) 4079 (4) 455 (5) None of these
Directions (Qs. 41-50): In each of the following questions a number series is given. A number in the
series is suppressed by letter 'A'. You have to find out the number in the place of 'A' and use this
number to find out the value in the place of the question mark in the equation following the series.
41. 36 216 64.8 388.8 A 699.84 209.952
A ÷36 = ?
(1) 61.39 (2) 0.324 (3) 3.24 (4) 6.139 (5) 32.4
42. 42 62 92 132 A 242 312
A + 14 = ? x 14
6 5
(1) 11 (2) 14 (3) 12 (4) 12 1 2 (5) 12 1 6
7 7
43. 4 7 12 19 28 A 52
A2 4 ?
(1) 1365 (2) 1353 (3) 1505 (4) 1435 (5) 1517
44. 18 24 A 51 72 98 129
3 4
A ?
7 5
23 12 2 2
(1) 12 (2) 11 (3) 12 (4) 14 (5) 10
35 35 5 7
3 3 9 9 27 27
45. A
8 4 16 8 32 16
A ?
3 6 6 3 9
(1) (2) (3) (4) (5)
2 8 4 4 8
46. 3 6 7 9 11 13 A 18
2A + 5 = ?
(1) 35 (2) 30 (3) 37 (4) 25 (5) 15
47. 5 8 13 A 29 40 53
A2 2 A ?
(1) 380 (2) 400 (3) 440 (4) 360 (5) 200
48. 1 A 4 12 15 60 64
A2 3 A 7 ?
(1) 2 (2) 17 (3) 12 (4) 20 (5) 8
49. A 27 38 51 66 83
A2 A3 ?
(1) 40 (2) 27 (3) 30 (4) 18 (5) 15
50. 2 5 9 14 A 27
4A
5A ?
2
(1) 60 (2) 20 (3) 100 (4) 70 (5) 50
Answers:
1. (1) 13 2 , 112 , 9 2 , 7 2
2. (5) 3 2, 2 3, 3 2, 2 3,
3. (4) 6, 2.5, 6, 2.5,
4. (2) A three tier sereis
5. (3) 5 52 , 5 52 ,
6. (1) 128
. , 0.96, 128
. , 0.96,
7. (2) 2 1, 3 1, 4 1, 5 1,
8. (5) 15
. , 2, 2.5, 3, 35
. ,
9. (1) 192, 96, 48, 24, 12, 6,
10. (2) 13 , 22 , 3 3 , 4 2 , 5 3 ,
11. (5) 6 12 , 5 2 2 , 4 32 , 3 4 2
12. (2) 112 , 13 2 , 15 2 , 17 2 ,
Replace (2) with (5)
2
13. (3) 1 13 , 2 2 2 , 3 32 , 4 4 2 ,
Replace (3) with (4)
14. (1) 12, 11, 10, 9
Replace (1) with (5)
15. (2) 1 1, 2 4, 3 7, 4 10,
Replace (2) with (4)
16. (2) 8, 7, 6, 5,
Replace (2) with (3)
17. (4) 12 1, 2 2 21, 32 3,
Replace (4) with (5)
18. (1) 27 2 , 25 2 , 23 2 , 212 ,
Replace (1) with (4)
19. (5) 1 0.25 12 , 22 0.25 2 2 , 3 0.25 32
20. (2) 6 ! 6, 5 ! 5, 4 ! 4, 3 ! 3
Replace (2) with (5)
21. (5)
22. (2) 4, 9, 25, 49
27,
23. (5) 9, 8, 7, 6,
24. (2) 2 3, 2 3, 2 3, 2 3,
25. (2) 23 2 , 19 2 , 15 2 , 112
26. (5) 1 3, 2 5, 3 7, 4 9,
27. (2) 6 6, 7 7, 8 8,
28. (1) 32, 22, 14, 8, 4,
29. (1) 3 7, 3 7,
30. (4) 3 2, 2 3, 3 2, 2 3,
31. (3) 1 2, 2 2, 3 2, 4 2,
32. (3) 1 1, 2 2, 3 3,
33. (4) 14 2 , 12 2 , 10 2 , 8 2 , 6 2 ,
34. (5) 0.5, 15
. , 2.5,3.5,
35. (3)
36. (3) 2, 3, 4, 5,
37. (2) 2 1, 2 1, 2 1, 2 1
38. (1) 22 , 4 2 , 6 2 , 8 2 ,
39. (1) 2 7, 2 7, 2 7, 2 7,
40. (4) 41 (3) 42 (2) 43. (5) 44. (1) 45. (5)
46. (1) 47. (4) 48. (2) 49. (3) 50. (1)
