Chapter 3
Series
A series or sequence consists of several terms. In other words, the unit of a sequence are called TERMS. Each term in the series has its own importance as there exists certain relationship between the two consecutive / alternating terms. This relationship is repeated in the series and based on this relationship; we are required to find out the missing term. We come across several types of questions based on any given series. (Base on whether the series consist of the alphabets or the numerals (numbers) or the words, the series can be classified into the following types).
1) NUMBER SERIES
2) ALPHABET SERIES
3) MIXED SERIES
It is quite easy to decipher an alphabetic series as it is easy to remember the place of alphabet in the series. In case of number series also the pattern can be found out but in case of mixed series or jumbled series, it is very difficult to remember the position of each term.
A mixed series comprises letters, numbers and symbols and unlike the English alphabet series, the number of terms is not fixed in such series. A mixed series may contain any number of terms viz. 23, 24, 26, 28, 30, 31 or 32. Such questions require sufficient practice as there are no definite SHORT CUTS for these questions. The questions on mixed series can be divided into two major heads:
i. Series of letters and numbers.
ii. Series of letters, numbers and symbols.
In questions based on mixed series one is required to judge and find out the relationship between the given terms and find out the answer. To begin with, count the terms and find out the relationship between the required terms
NUMERICAL SERIES
ANALYSIS OF NUMERICAL SERIES: A close analysis of the above examples show that the number series can be of the following types.
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, â€¦â€¦. The difference between two consecutive numbers is 2. So it makes a series.
Ex: ii) 1, 2, 4, 7, 11, 16â€¦â€¦. 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 1^{2}, 2^{2}, 3^{3}, 4^{2}, 5^{2},
Hence 6^{2}^{} = 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 : 1^{3} â€“ 1, 2^{3} â€“ 2, 3^{3} â€“ 3, 4^{3} â€“ 4, 5^{3} â€“ 5, 6^{3} â€“ 6.
Next number = 7^{3} â€“ 7 = 343 â€“ 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.
ALPHABET SERIES
In problems based on alphabet series, the pattern of alphabet in the series is to be deducted and the next term is to be found out. There are no set rules, yet the problems can easily be solved if the place number of each alphabet is memorized. Like for Kâ€™s place value in the alphabet sequence is 11 and that of V is 22 , that of Q is 17 and X is 24. If these places are known, the questions become easy to solve.
There can be omission of alphabets one each time. Alphabets may be omitted in an increasing order or decreasing order; for example one each time or two each time or three each time and so on. There can also be alternate order such as first one alphabet is skipped, then two may be skipped and then three may be omitted. The skipping pattern may be
Regular Order: In this case the number of alphabets skipped remain the same through the series.
Increasing Order: Each time the number of alphabets skipped increase in a given pattern.
The theory can only be understood with the help of practical examples only.
Ex. 1. What terms will fill the blank spaces?
A, B, C , D, E, (â€¦â€¦), (â€¦â€¦) 1) O, K 2) N, M 3) K, S 4) F, G
Sol. : The given series consist of alphabets in their original order. So, the missing terms would be F and G. Ans. (4)
Ex. 2. What terms will fill the blank spaces?
Z, X, V, T, R, (â€¦â€¦), (â€¦â€¦)
1) O, K 2) N, M 3) K, S 4) P, N
Sol. : Clearly, the given series consist of alternate letters in a reverse order. The alphabets are skipped one at a time. So, the missing terms would be P and N. Ans. (4)
Ex. 3 Find the missing alphabet: B, D, G, K, P, (â€¦â€¦), (â€¦â€¦)
Sol. : Each time the number of letters skipped increases by one. So the answer here is V.
B | D | G | K | P | V |
2 | 4 | 7 | 11 | 16 | 22 |
B is second alphabet in the series, D is fourth in the series, G is seventh, K is eleventh, P is sixteenth and so the next will be twenty second i.e. V of the series.
Ex.4
OTE, PUF, QVG, RWH, (â€¦â€¦â€¦.)
1) SYJ 2) TXI 3) SXJ 4) SXI
Sol. The first letters of the terms are in alphabetical order, and so are second and third letters.
OPQRS, TUVWX, EFGHI
Ans. SXI (4)
Ex. 5 FLP, INS, LPV, (â€¦â€¦)
(a) ORY (b) UXZ (c) VXY (d) SVW
The first and third letters of each term are moved three steps forward and the second letter is moved two steps forward to obtain the corresponding letters of the next term.
We pick one letter from one group at a time. Like F from FLP, I from INS, L from LPV.
F_{6} I_{9} L_{12} O_{15}
L_{12} N_{14 } P_{16 } R_{18}
P_{16 } S_{19} V_{22} Y_{25}
MIXED SERIES
Series Consisting of Letters & Numbers
Directions (1 â€“ 3): Study the following arrangement carefully and answer the questions given below:
W 1 5 E J R 2 M A 9 T K U N 4 B I 8 D H 3 F 6 P Z 7 Q
Ex. 1. How many such consonants are there, each of which is immediately preceded by a number and immediately followed by a vowel?
(1) None
(2) One
(3) Two
(4) Three
Ex. 2. Which of the following is the seventh to the left of 4?
(1) M
(2) A
(3) F
(4) 3
Ex. 3. How many such vowels are there, each of which is immediately preceded and also immediately followed by a consonant?
(1) None
(2) One
(3) Two
(4) Three
ANSWER WITH EXPLANATION
1. (3)
2. (1)
3. (2)
SERIES CONSISTING OF LETTERS, NUMBERS & SYMBOLS
In order to make series more complicated, letters and numbers are combined with different symbols. Consider the following examples:
Directions (4 â€“6): Study the following arrangement carefully and answer the questions given below :
K P 5 # 7 M N E 2 D A Â¶ 4 F H I T 9 1 $ U 6% W 3
Ex. 4: How many such vowels are there in the above arrangement, each of which is either immediately preceded by a number or immediately followed by a symbol?
(1) None
(2) One
(3) Two
(4) Three
Ex. 5: Which of the following is the sixth to the right of the fifteenth element from right end in the above arrangement?
(1) T
(2) 7
(3) 2
(4) U
Ex. 6: Which of the following is in exactly middle between seventh from the left and ninth from the right end?
(1) A
(2) H
(3) 4
(4) None of these
ANSWER WITH EXPLANATION
4. (2)
5. (1)
15th element from right Ãž A
6th to right of A Ãž T
Trick Required element
=15 â€“ 6 = 9th term from right.
6. (4) None of these
MISCELLANEOUS TYPE
Ex. 12. How may such digits are there in the number 82134967 each of which is as far away from the beginning of the number as when they are arranged in descending order?
(1) None
(2) One
(3) Two
(4) Three
Answer: (2)
4 is such number in the series which is at the same position in the original and also when arranged in a descending order.
Ex. 13. How may such digits are there in the number 57394162 each of which is as far away from the beginning of the number as when they are arranged in descending order?
(1) None
(2) One
(3) Two
(4) Three
Answer: (3) Two. The figure 7 & 4 are at second and fifth position from left. When we place these figures in descending order these will again appear at second and fifth position from left. So our answer is ( 3 ) i.e. there is one such digit in the series which is at same position after arrangement in descending order as when it was in the original series.
7 and 4 are such numbers in the series which is at the same position in the original and also when arranged in a descending order.