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Worked Examples

Example-1
2 5 8 11 14 17 ___.
Solution
In this problem, each term is obtained by adding 3 to its previous term.
 
In other words, the difference between any two consecutive terms remains constant and it is 3.
 
In shortcut, the pattern is indicated as under.
 
 
23369.png
 
Answer: 20
 
 
Example-2
3 4 6 9 13 18 __.
Solution
In this problem, the difference between two consecutive terms is constantly increasing by 1.
 
In shortcut, the pattern is indicated as under.
 
23378.png

Note: From now onwards, we use only shortcut method to explain the pattern followed.

Answer: 24
 
 
Example-3
4 8 16 32 64 __.
Solution
23430.png
 
Answer: 128
 
 
Example-4
729 243 81 27 __ 3.
Solution
23920.png
 
Answer: 9
 
 
Example-5
2 5 11 23 47 95 __.
Solution
23955.png
 
This problem can also be solved by an alternate method.
 
23947.png

In the above pattern, the difference between two consecutive terms is doubling in each step.
 
From the above, it is clear that the same problem can be tackled in two different methods.
 
Hence, different patterns can be identified for the same problem in certain cases.
 
Answer: 191
 
 
Example-6
1 2 5 16 65 __.
Solution
23974.png
 
Answer: 326
 
 
Example-7
1 2 6 21 88 __.
Solution
23990.png
 
Answer: 445
 
 
Example-8
126 62 30 14 __ 2.
Solution
24001.png
 
Answer: 6
 
 
Example-9
2 3 5 7 11 13 __.
Solution
In this problem, the terms are consecutive prime numbers.
 
The next prime after 13 is 17.
 
Hence, the missing term is 17.
 
Answer: 17
 
 
Example-10
2 5 11 17 23 31 __.
Solution
In this problem, the terms are alternate prime numbers.
 
37 and 41 are the next two prime numbers after 31.
 
By skipping 37, we should take 41 as the correct missing number.
 
Answer: 41
 
 
Example-11
1 4 9 16 25 __.
Solution
All the terms in the above problem are perfect squares of natural numbers.
 
Hence, the missing number is 36.
 
In shortcut method, the pattern is indicated as under:
 
1 4 9 16 25 36
12 22 32 42 52 62

Answer: 36
 
 
Example-12
5 10 17 26 __ 50.
Solution
5 10 17 26 37 50
22+1 32+1 42+1 52+1 62+1 72+1

Answer: 37
 
 
Example-13
2 6 12 20 30 42 __.
Solution
2 6 12 20 30 42 56
12+1 22+2 32+3 42+4 52+5 62+6 72+7

The above problem can be tackled by a few different methods also.

Alternate method 1:
 
24574.png

Alternate method 2:
 
2 6 12 20 30 42 56
1 × 2 2 × 3 3 × 4 4 × 5 5 × 6 6 × 7 7 × 8

Alternate method 3:
 
2 6 12 20 30 42 56
22 - 2 32 - 3 42 - 4 52 - 5 62 - 6 72 - 7 82 - 8

In all the above three methods, we get 56 as the correct answer.
 
Answer: 56
 
 
Example-14
2 9 28 65 126 __.
Solution
2 9 28 65 126 217
13+1 23+1 33+1 43+1 53+1 63+1

Answer: 217
 
 
Example-15
2 12 36 80 150 __.
Solution
2 12 36 80 150 252
13+12 23+22 33+32 43+42 53+52 63+62

Answer: 252
 
 
Example-16
11 13 17 25 32 __ 47.
Solution
This is a tricky problem.
 
If we analyse the first four terms, it appears as if the difference between two consecutive terms keeps on doubling in each stage.
 
However, this pattern does not hold between the two consecutive terms 25 and 32.
 
Hence, we should think of some different pattern.
 
Take any term.
 
Find the sum of digits of the term and add it to the same term.
 
We get the next term.
 
This pattern continues throughout the series.
 
By applying this pattern, we can find the missing term as 37.
 
 
25064.png
 
Answer: 37
 
 
Example-17
13 16 22 26 38 62 __ 102.
Solution
In this problem, product of the digits of a term is added to the term to get the next term.
 
 
25076.png
 
Answer: 74
 
 
Example-18
4 5 9 18 34 __ 95.
Solution
25092.png
 
Answer: 59
 
 
Example-19
2 3 4 6 12 __ 156.
Solution
25249.png
 
Answer: 36
 
 
Example-20
2 3 10 15 26 35 __.
Solution
2 3 10 15 26 35 50
12+1 22–1 32+1 42–1 52+1 62–1 72+1

Answer: 50
 
 
Example-21
2 6 15 64 315 __.
Solution
25308.png
 
Answer: 1896
 
 
Example-22
1 2 3 3 5 5 7 7 9 11 __ __.
Solution
We could not notice any common pattern in the above series.
 
On close observation, we notice that it is a combination of two series.
 
We shall take the terms only at odd-numbered positions and analyse the pattern first.
 
 
25317.png

Now, we shall take the numbers at even-numbered positions and analyse the pattern.
2 3 5 7 11 __
 
The terms are consecutive prime numbers.
 
Hence, the missing number in this series is 13, which is the next prime number.
 
This type of series is called twin series or combination/mixed series.
 
Answer: 11, 13
 
 
Example-23
1 2 2 3 6 5 24 8 120 __.
Solution
This problem is again a combination series.
 
Let us take the alternate terms and analyse the pattern first.
 
The terms at odd-numbered positions follow the pattern given as follows:
 
 
25393.png

The terms at even-numbered positions can be analysed as under:
 
25384.png

Hence, the missing number is 12.
 
Answer: 12
 
 
Example-24
4 12 15 60 64 320 325 __.
Solution
25406.png
 
Answer: 1950
 
 
Example-25
61 52 63 94 46 __.
Solution
This is a tricky problem. Closely observe each term of the series.
 
On interchanging the digits of each term, we get a number which is a perfect square of a natural number.
 
The first term is obtained by interchanging the digits of 42.
 
The second term is obtained by interchanging the digits of 52 and so on.
 
So, we can get the missing term by interchanging the digits of 92.
 
Hence, 18 is the missing number.
 
Answer: 18
 




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