## INTRODUCTION

Questions on series are now extremely common in all the Prelims and Mains exams. They are generally of three types – number, letter and alphanumeric – but the most commonly asked questions are on number series.

**NUMBER SERIES**

You are given a sequence of numbers that follow a common pattern. You need to understand this pattern and either identify the next term (more common) or find a missing term. There are infinite number of ways to create a number series by using one or more patterns. However, with practice you can start identifying the more common patterns such as difference, product, ratio, squares, cubes, powers factorials etc. Some basic series are:

Even Numbers: 2, 4, 6, 8…

Odd Numbers: 1, 3, 5, 7, 9 …

Prime Numbers: 2, 3, 5, 7, 11, 13, 17 …

Composite Numbers: 4, 6, 8, 9, 10, 12, 14, 15 …

Squares: 1, 4, 9, 16, 25, 36, 49, ….

Cubes: 1, 8, 27, 64, 125, 216, …

Factorials: 1, 2, 6, 24, 120, 720, ….

Note that ‘1’ is neither prime nor composite. Depending on the numbers used, a series can be increasing, decreasing or alternate. The common patterns used to create number series are:

**DIFFERENCE BASED**

This is the most basic and common form of series. If possible, this is the first thing you should check when you try to find a pattern in the series. You should take the difference between consecutive terms of a series. This difference can be constant (e.g. +5, −3 etc) or have some pattern of its own. This can be one of the patterns explained above e.g. if the series is: 32 33 41 68 132, take the difference to get 1 8 27 64. Since these numbers are clearly a series of perfect cubes, the next number in this sequence is 125. Hence, the next number in the original series is 132 + 125 = 257. Occasionally, you might not get a pattern after a first level of difference. Ten, you should take a difference of these differences and check.

Directions for examples 1 to 5: Find the missing term to complete the given series.

Example 1:

58, 70, 84, 100, ___

(1) 112(2) 119(3) 116(4) 118(5) 122

Solution:

Consider the difference between consecutive terms: 12, 14, 16, …

Observe that the difference is a series of consecutive even numbers.

Hence, next difference = 18 and next number = 100 + 18 = 118

Hence, option 4.

Example 2:

3, 9, 18, 30, ___

(1) 33(2) 36(3) 45(4) 39(5) 42

Solution:

This is an example of multiple patterns applying to the same set of numbers. In each case, the answer turns out to be the same.

Logic 1: Take the difference of consecutive terms i.e. 6, 9, 12, ….

The difference is a series of consecutive multiples of 3.

Hence, next difference = 15 and next term = 30 + 15 = 45

Hence, option 3.

Logic 2: Each term can be written as:

3 = 3 × 1; 9 = 3 × 3; 18 = 3 × 6; 30 = 3 × 10

Observe that the multiples of 3 in the above series themselves form a series 1, 3, 6, 10. The difference between consecutive terms of this new series is 2, 3, 4 and so on.

Hence, the next difference should be 10 + 5 = 15. Thus, the required multiple of 3 is 15.

Required term = 3 × 15 = 45

Hence, option 3.

Logic 3:

3 = 3 × 1

9 = 3 + 3 × 2

18 = 9 + 3 × 3

30 = 18 + 3 × 4

The pattern is (Previous term) + (3 × Position of term in the sequence)

Therefore, the next term should be 30 + 3 × 5 = 45

Hence, option 3.

Note:

Though the answer can be obtained through various patterns, you should always select the pattern that you first think of in the exam. Sometimes, the different patterns possible give different answers. In that case, you should look at the options and select the pattern whose answer is given in the options.

Example 3:

336, 305, 268, 227, 184, ___

(1) 137 (2) 163(3) 146(4) 133 (5) 129

Solution:

Consider the difference between consecutive terms: −31, −37, −41, −43.

If you ignore the minus sign, the differences form a series of consecutive primes.

Since the next prime is 47, the required difference = −47.

Hence, required term = 184 – 47 = 137

Hence, option 1.

Example 4:

1, 1, 2, 3, 5, 8, ___

(1) 9(2) 10(3) 12(4) 13(5) 15

Solution:

Observe that, starting from the third term, each term is the sum of the 2 terms immediately preceding it.

Hence, the required term = 5 + 8 = 13.

Hence, option 4.

Note:

The series given in this example is a special series known as the Fibonacci Series in which the sum of two successive terms is the next term.

**PRODUCT BASED**

Just like difference based series, product based series either have a common ratio between terms or the ratio also shows some pattern. Common patterns are powers, factorials and multiples. A common feature of such series is that the value of consecutive terms increases/decreases quite sharply. However, first level subtraction often helps in identifying the underlying pattern.

PRODUCT

The series may be based on simple application of factors or multiples.

Example 6:

Find the missing term. 2, 12, 30, 56, ___

(1) 77(2) 90(3) 79(4) 72(5) 92

Solution:

Each number can be expressed as a product of consecutive numbers:

1 × 2 = 2; 3 × 4 = 12; 5 × 6 = 30; 7 × 8 = 56

Hence, required term = 9 × 10 = 90

Hence, option 2.

Alternatively,

Consider the difference between consecutive terms: 10, 18, 26.

These differences form a series increasing by 8. Hence, next difference =

26 + 8 = 34.

Hence, the required term = 56 + 34 = 90.

Hence, option 2.

Example 7:

Find the missing term. 24, 12, 12, 18, 36, ___

(1) 90(2) 92(3) 78(4) 67(5) 77

Solution:

Consider the ratio between consecutive terms: 0.5, 1, 1.5, 2.

Since the ratio keeps increasing by 0.5, the next ratio is 2.5.

Hence, required term = 36 × 2.5 = 90

Hence, option 1.

POWER

These include questions where the pattern is related to squares, cubes or higher powers. Here, the value of the term increases even more sharply.

Example 8:

Find the missing term. 2, 6, 30, 260, ___

(1) 420(2) 500(3) 3140(4) 610(5) 3130

Solution:

The sharp increase in value indicates that the pattern may be based on powers.

The given series can be expressed as (11 + 1), (22 + 2), (33 + 3), (44 + 4)

Therefore, the required term = 55 + 5 = 3130

Hence, option 5.

Example 9:

Find the missing term. 11, 24, 39, 416, ___

(1) 626(2) 525(3) 552(4) 523(5) 5025

Solution:

Here, the numbers do not show an obvious pattern either using differences or simple products. If we split each number in 2 parts, we see that one part is the square of the other part.

Each term is of the form nn2.

Hence, the 5th term should be 552 i.e. 525.

Hence, option 2.

FACTORIAL

This involves the fastest growth in values since the factorials of the first 7 natural numbers are 1, 2, 6, 24, 120, 720 and 5040.

Example 10:

Find the missing term. 1, 1, 4, 36, ___

(1) 36(2) 51(3) 576(4) 81(5) 225

Solution:

The pattern seen in the above series is (n!)2.

Each term can be expressed as: (0!)2, (1!)2, (2!)2, (3!)2 and so on.

Therefore the required term should be (4!)2 = (24)2 = 576.

Hence, option 3.

Consider ratio between consecutive terms: 1, 4, 9. These are squares of consecutive natural numbers.

Hence, next ratio = 16 and required term = 36 × 16 = 576

Hence, option 3.

**ALTERNATING SERIES**** **

An alternating series is a combination of two or more series. Each series can have different patterns applied to it and then combined to form a series. In a combination of 2 series, alternate terms follow the same pattern. An alternating series can be a combination of more than 2 series as well. If you are asked to find two or more values, it is very likely to be an alternating series.

Example 11:

Find the missing term. 0, 3, 3, 4, 6, 5, 9, 6, ___

(1) 9(2) 5(3) 4(4) 10(5) 12

Solution:

Since the values increase and decrease, you should check for alternate series.

Taking alternate terms together, we get (0, 3, 6, 9, ?) and (3, 4, 5, 6). One is a series of consecutive multiples of 3 while the other is a series of consecutive integers.

Hence, the required number is the next multiple of 3 i.e. 12.

Hence, option 5.

Example 12:

Find the missing term. 1, 2, 7, 12, 21, 70, 43, _____

(1) 124(2) 224(3) 184(4) 150(5) 212

Solution:

Since the values increase and decrease, you should check for alternate series.

Taking alternate terms together, we get (1, 7, 21, 43) and (2, 12, 70, ?).

The first series is of the form [n + (n – 1)2]; where n is the series of consecutive odd numbers starting from 1.

The second series is of the form [n + (n – 2)3]; where n is the series of even numbers starting from 2.

The required number is part of the second series.

Hence, required number = 8 + (8 – 2)3 = 8 + 63 = 8 + 216 = 224

Hence, option 2.

**MISCELLANEOUS**** **

These can involve a combination of patterns or series and cannot be directly classified.

Example 13:

Find the missing term. 10, 103, 18, 187, ___

(1) 979(2) 26(3) 9(4) 251(5) 34

Solution:

The difference of consecutive terms does not form a logical series.

Here, look at the actual digits of each term. The sum of digits of each term is:

10: 1 + 0 = 1

103: 1 + 0 + 3 = 4

18: 1 + 8 = 9

187: 1 + 8 + 7 = 16

Thus, the sum of digits for each term is the square of consecutive natural numbers.

Hence, the next term should have sum of digits = 52 = 25.

Only 979 satisfies this condition.

Hence, option 1.

Example 14:

Find the missing term. 0, 1, 0, 8, 2, 7, 6, ___

(1) 2(2) 3(3) 4(4) 5(5) 8

Solution:

It is a series of cubes of natural numbers but each digit of the number is written as a separate term. Here, each cube has been expressed as a two digit number split into two parts i.e. 1 as 01, 8 as 08 and so on.

The next cube is 43 = 64. Since ‘6’ is already present, the next term = 4.

Hence, option 3.

**Tips: **

- For solving problems on numbers series, first observe the difference between the numbers. The difference may be constant or may form a pattern.

- If the first level of subtraction does not show a pattern, subtract the differences from each other again. This may show a pattern. Continue till a consistent pattern is found.

- If the difference between consecutive terms is very large and there is no constant pattern in the difference, it may be a product series. In a product series, the terms increase/decrease at a greater rate compared to a difference based series. Multiplicative series are generally based on pure multiplication, powers or factorials.

- First check if the terms show some relationship with the multiple or factor of some number or group of numbers (such as even multiples of prime numbers).

- If no factor/multiple based relationship is found, check if the numbers lie close to squares or cubes or higher powers of any number.

- Remember the factorials of the first 7-8 numbers and see if the terms lie close to the factorials.

- If the terms increase and decrease alternately, it may be an alternate series with two different patterns. If a question has more than 1 blank, in most cases it implies a combination of 2 or more series. If nothing works out, check the relationship between the digits of the terms, especially if some terms are very small in value and some are very large or if the terms seem to be random.

- If the middle term is missing, check the pattern between the 1
^{st}and 2^{nd}term, as well as the 4^{th}and 5^{th}term to see if there is an alternating pattern. Then use the answer option to see which alternative suits the best.

- In letter series, always write down the position of the letters in the alphabet and then find the relationship.

- In alphanumeric as well as letter series, focus on one element at a time e.g. first letter, first number. Based on its pattern, you may be able to eliminate some options.

- If you are unable to get the pattern for a series based question in approximately one minute, leave it for the time being. Do not spend too much time on it.

- If two patterns are visible for a given series, check if the answer by both is the same or different. If it is different, check which answer is given in the options. If very rare cases where both are given, you can either of the two as the answer.