## Thursday, 16 February 2017

### Series

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:

i) Addition/Subtraction – 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)

#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 1­­­­1 numbers are added in the form of nn by adding 4 i.e. 22, 27 i.e. 33, 256 i.e. 44. Next number to add should be 3125 i.e. 55 which will make the next term (288 + 3125) = 3413.

ii) Multiplication/Division - 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)

#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.

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

#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

iv) Square/Cube Numbers – These are any patterns related with square and cube numbers.

#8. 1, 2, 6, 15, 31, ? – Here consecutive square numbers are being added to each term. So after adding 42 i.e. 16 next term should be added with 52 i.e. 25 which will give us 56.

#9. 9, 28, 65, 126, 217, ? – Starting with 23+1, consecutive cube numbers have been listed after adding 1 to each. Next term should be 73+1 = 344.

v) Sum of Digits/Difference of Digits – When next term is derived by adding the digits of previous term or finding the difference between the digits of previous term.

#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.

vi) Mix Series – When along with addition/subtraction, division/multiplication is mixed together it leads to another pattern.

#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.

vii) Alternate Series – Alternate series is when more than one series is mixed together.

#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 1st, 4th, 7th… terms are being multiplied with 3 (3, 9, 27, 81); the 2nd, 5th, 8th …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. 12th 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:

i) Vowels/Consonants: 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.

#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.

ii) Convert in numbers: 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.

iii) Opposite letters: Any pair of letters whose position can be interchanged when counted from A-Z and reverse Z-A are called Opposite letters.

Eg. A is first letter from beginning and 26th from the end; so its opposite shall be Z which is 26th from the beginning and first from the end. Similarly J is 10th from the beginning and 17th from the end; so its opposite shall be Q which is 17th if counted from A-Z and 10th 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

iv) Corresponding letters: Any pair of letters wherein the interval is consistent between them, both when counted from front or backwards are called 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 12), D (4 i.e. 22), I (9 i.e. 32), P (16 i.e. 42). Hence, the first letter of the next pair shall be position 25 i.e. 52 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 –

#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)

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 %H7 i.e. option c)