Series is one of the most traditional topics of
Reasoning and also usually the most familiar one even for a first timer at
competitive and aptitude tests. While some of the popular management entrance
tests have curtailed the number of questions from this chapter in recent times,
it will not be advisable to ignore this topic, for it could mean missing out on
some crucial score which could affect one’s overall percentile and prove costly
especially if it’s anywhere near to the cut off.

There is not much to offer in this topic conceptually;
however it is important to have an approach in mind while taking on its
questions. Questions from Series are mostly based on one or more of the
following patterns:

**– Here numbers are added or subtracted in a pattern. Usually terms are relatively close to each other in such patterns; however there are exceptions (eg #3)**

*i) Addition/Subtraction*
#1. 3, 4, 7, 12, ? – Here the series begins with 3
and increasing order of odd numbers are being added. After adding 1, 3, 5, it
is now turn for adding 7 which will give the answer 19.

#2. 31, 38, 33, 40, 35, ? – Here the numbers 7 and 5
are added and subtracted alternately. After subtracting 5 from 40 and making it
35, it is now turn to add 7 and the answer shall be 42.

#3. 1, 5, 32, 288, ? – Beginning with 1

_{}^{1}numbers are added in the form of n^{n}by adding 4 i.e. 2^{2}, 27 i.e. 3^{3}, 256 i.e. 4^{4}. Next number to add should be 3125 i.e. 5^{5}which will make the next term (288 + 3125) = 3413.**- Where certain numbers are multiplied or divided in a pattern. Mostly this pattern has terms far apart from each other to allow the scope of multiplication/division; however there are exceptions (eg #5)**

*ii) Multiplication/Division*
#4. 362880, 45360, 7560, 1890, ? – Here the numbers
are being divided by decreasing consecutive even numbers starting with 8
followed by 6 and 4. Next divisor should be 2 which will result in 945.

#5. 8, 4, 4, 6, 12,? – Here the first term has been
multiplied with 0.5 and then subsequent terms have been multiplied with numbers
at an interval of 0.5 i.e. 1, 1.5 and 2. Next term should hence be multiplied
with 2.5 giving a figure of 30.

**– These are patterns pertaining to prime numbers and may also be combined with above mentioned patterns (like addition in #7)**

*iii) Prime Numbers*
#6. 97, 79, 67, 53, ? – Here prime numbers have been
listed in decreasing order skipping two prime numbers in between. So after 97,
89 & 83 have been skipped and similarly 73, 71, 61, 59 have been skipped
till 53. Similarly after 53, 47 & 43 should be skipped to result in 41 as
the next term.

#7. 1, 3, 6, 11, 18, ? – Here 2, 3, 5 and 7 have
been added on the 4 terms respectively. As these are consecutive prime numbers,
the next number to add should be 11 which is the next prime number after 7 and
shall result in 29 as the next term

**– These are any patterns related with square and cube numbers.**

*iv) Square/Cube Numbers*
#8. 1, 2, 6, 15, 31, ? – Here consecutive square
numbers are being added to each term. So after adding 4

^{2}i.e. 16 next term should be added with 5^{2}i.e. 25 which will give us 56.
#9. 9, 28, 65, 126, 217, ? – Starting with 2

^{3}+1, consecutive cube numbers have been listed after adding 1 to each. Next term should be 7^{3}+1 = 344.**– When next term is derived by adding the digits of previous term or finding the difference between the digits of previous term.**

*v) Sum of Digits/Difference of Digits*
#10. 24, 30, 33, 39, 51, ? – Here the numbers added
on the terms are a sum of the digits of the previous term: 24 + (2+4), 30 +
(3+0), 33 + (3+3) and so on. Hence next term should be 51 + (5+1) = 57.

#11. 29, 36, 39, 45, 46, ? – Here the numbers added
on the terms are a difference of the digits of the previous term: 29 + (9-2),
36 + (6-3), 39 + (9-3), 45 + (5-4). Hence the next term should be 46 + (6-4) =
48.

#12. 3, 7, 10, 17, 27, 44, ? – Here third term
onwards the series is the sum of the previous two terms; so the next term
should be (27 + 44) = 71.

**– When along with addition/subtraction, division/multiplication is mixed together it leads to another pattern.**

*vi) Mix Series*
#13. 2, 7, 17, 37, 77, ? – Here the terms are being
multiplied with 2 and then 3 is added to them. (2 x 2) + 3 = 7, (7 x 2) + 3 =
17, (17 x 2) + 3, and so on. Hence the next term should be (77 x 2) + 3 = 157.

**– Alternate series is when more than one series is mixed together.**

*vii) Alternate Series*
#14. 1, 2, 3, 3, 5, 5, 7, 7, 9, 11, 11, 13, ? – Here
the odd and even positions are two different series running alternately. The
odd positions are a series of consecutive odd numbers, while the even positions
are a series of prime numbers. Hence the next term will be the next prime
number after 11 i.e. 13.

#15. 3, 10, 16, 9, 20, 8, 27, 30, 4, 81, 40, ? –
Here 3 different series have been mixed together such that 1

^{st}, 4^{th}, 7^{th}… terms are being multiplied with 3 (3, 9, 27, 81); the 2^{nd}, 5^{th}, 8^{th}…terms are being added with 10 (10, 20, 30, 40); and every third term is being divided by 2 (16, 8, 4). Hence the next term i.e. 12^{th}term should be 4 divided by 2 i.e. 2.####
**Letter
Series**

The following points need to be kept in mind while
solving letter series:

**Unlike numbers, alphabet letters do not have too many properties. The most intrinsic property is the very nature of the letter i.e. vowel or consonant. It can be identified immediately if the series is based on this pattern.**

*i) Vowels/Consonants:*
#16. B, F, J, P, ? – Here the consonants following
the vowels have been arranged in ascending order of positions. Hence after P,
the next vowel is U which will be followed by V and that will be our answer.

**When a series is given in letter form and there is no apparent pattern with vowel/consonants, then it is advisable to convert the letters into their position number in the alphabet and treat it as number series and apply all that has been mentioned above.**

*ii) Convert in numbers:***Any pair of letters whose position can be interchanged when counted from A-Z and reverse Z-A are called Opposite letters.**

*iii) Opposite letters:*
Eg. A is first letter from beginning and 26

^{th}from the end; so its opposite shall be Z which is 26^{th}from the beginning and first from the end. Similarly J is 10^{th}from the beginning and 17^{th}from the end; so its opposite shall be Q which is 17^{th}if counted from A-Z and 10^{th}when considered from Z-A.

__An easy way to identify such opposite pairs is to find the sum of their positions from A onwards and the total shall always be 27.__
#17. EV, JQ, OL, TG, ?? – Here all are pairs of
opposite letters where the first letter in each pair is at an interval of 5 (based on position), 10,
15, 20. As the interval between first letter of consecutive pairs is 5, first
letter of the next pair shall be (20 + 5) = 25 i.e. Y and its opposite letter
shall be (27 – 25) = 2 i.e. B

*Any pair of letters wherein the interval is consistent between them, both when counted from front or backwards are called Corresponding letters.*

**iv) Corresponding letters:**
Eg. When counting from A to N, there is an interval
of 13 and if we continue forward and come back to A after Z, then again the
interval will be of 13. Similarly we can take a pair like F-S which when
counted from F to S or from S to F gives us the same result.

__A simple way to identify such pairs will be to verify an interval of 13 between the two letters.__

#18. AN, DQ, IV, PC, ?? – Each pair here are at an
interval of 13 and hence are corresponding letters. Also the first letters in
every pair are at positions which are consecutive perfect squares A (1 i.e 1

^{2}), D (4 i.e. 2^{2}), I (9 i.e. 3^{2}), P (16 i.e. 4^{2}). Hence, the first letter of the next pair shall be position 25 i.e. 5^{2}which will be Y and it will be paired with (25 – 13) = 12 i.e. L which means the next pair shall be YL.
The chapter of Series can be mastered by practice alone,
as one gets familiar with more and more patterns. However it is important to
have an approach in mind so that one doesn’t forget to try any of the patterns.

The chapter extends to influence some other popular
topics like Odd Man Out and Analogies which have been discussed on another post of this
blog:

Coding & Decoding and Input-Output also have
certain questions based on the concepts of Series. Following is an example –

Hence, from the first
and fourth sentence it can be understood that the first letter ‘T’ is denoted
by the symbol ‘%’ and as Triumph is a seven letter word ending with H, its code
will be

#19. In a certain code language if,

‘Put on your thinking cap’ is written as #N2, ?P3,
%G8, @T3, *R4

‘Life is not easy’ is written as {Y4, !E4, ]S2, ^T3

‘It cannot be helped’ is written as &E2, +D6,
?T6, ]T2

‘If you miss the obvious’ is written as ]F2, %E3,
#S7, =S4, *U3

What will be the code for ‘Triumph’?

a) ^T6 b)
]M8 c) %H7 d) None of these

- Here the codes of words have been jumbled and can
be identified as per following pattern:

i) The first letter has been assigned a unique
symbol (eg. ‘I’ as ‘]’ can be checked in second, third and fourth statements)

ii) The letter in the code represents the last
letter of the word (eg. the last letters of the words in first statement are T,
N, R, G, P which are also in the codes)

iii) The number in the code represents the number of
letters in the word (eg. the number of letters of words in second statement are
4, 2, 3, 4 which are also in the respective codes)

**%H7 i.e. option c)**
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