# 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.

**Answer:**20Example-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.

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

**Note:****Answer:**24

Example-3

4 8 16 32 64 __.

Solution

**Answer:**128

Example-4

729 243 81 27 __ 3.

Solution

**Answer:**9

Example-5

2 5 11 23 47 95 __.

Solution

This problem can also be solved by an alternate method.

In the above pattern, the difference between two consecutive terms is doubling in each step.

**Answer:**191

Example-6

1 2 5 16 65 __.

Solution

**Answer:**326

Example-7

1 2 6 21 88 __.

Solution

**Answer:**445

Example-8

126 62 30 14 __ 2.

Solution

**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:**17Example-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:**41Example-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 |

â†“ | â†“ | â†“ | â†“ | â†“ | â†“ |

1^{2} |
2^{2} |
3^{2} |
4^{2} |
5^{2} |
6^{2} |

**Answer:**36Example-12

5 10 17 26 __ 50.

Solution

5 | 10 | 17 | 26 | 37 |
50 |

â†“ | â†“ | â†“ | â†“ | â†“ | â†“ |

2^{2}+1 |
3^{2}+1 |
4^{2}+1 |
5^{2}+1 |
6^{2}+1 |
7^{2}+1 |

**Answer:**37

Example-13

2 6 12 20 30 42 __.

Solution

2 | 6 | 12 | 20 | 30 | 42 | 56 |

â†“ | â†“ | â†“ | â†“ | â†“ | â†“ | â†“ |

1^{2}+1 |
2^{2}+2 |
3^{2}+3 |
4^{2}+4 |
5^{2}+5 |
6^{2}+6 |
7^{2}+7 |

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

**Alternate method 1:**

**Alternate method 2:**

2 | 6 | 12 | 20 | 30 | 42 | 56 |

â†“ | â†“ | â†“ | â†“ | â†“ | â†“ | â†“ |

1 Ã— 2 | 2 Ã— 3 | 3 Ã— 4 | 4 Ã— 5 | 5 Ã— 6 | 6 Ã— 7 | 7 Ã— 8 |

**Alternate method 3:**

2 | 6 | 12 | 20 | 30 | 42 | 56 |

â†“ | â†“ | â†“ | â†“ | â†“ | â†“ | â†“ |

2^{2} - 2 |
3^{2} - 3 |
4^{2} - 4 |
5^{2} - 5 |
6^{2} - 6 |
7^{2} - 7 |
8^{2} - 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 |

â†“ | â†“ | â†“ | â†“ | â†“ | â†“ |

1^{3}+1 |
2^{3}+1 |
3^{3}+1 |
4^{3}+1 |
5^{3}+1 |
6^{3}+1 |

**Answer:**217

Example-15

2 12 36 80 150 __.

Solution

2 | 12 | 36 | 80 | 150 | 252 |

â†“ | â†“ | â†“ | â†“ | â†“ | â†“ |

1^{3}+1^{2} |
2^{3}+2^{2} |
3^{3}+3^{2} |
4^{3}+4^{2} |
5^{3}+5^{2} |
6^{3}+6^{2} |

**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.

**Answer:**37Example-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.

**Answer:**74Example-18

4 5 9 18 34 __ 95.

Solution

**Answer:**59

Example-19

2 3 4 6 12 __ 156.

Solution

**Answer:**36

Example-20

2 3 10 15 26 35 __.

Solution

2 | 3 | 10 | 15 | 26 | 35 | 50 |

â†“ | â†“ | â†“ | â†“ | â†“ | â†“ | â†“ |

1^{2}+1 |
2^{2}â€“1 |
3^{2}+1 |
4^{2}â€“1 |
5^{2}+1 |
6^{2}â€“1 |
7^{2}+1 |

**Answer:**50

Example-21

2 6 15 64 315 __.

Solution

**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.

Now, we shall take the numbers at even-numbered positions and analyse the pattern.

2 3 5 7 11 __

*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:

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

Hence, the missing number is 12.

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

Hence, the missing number is 12.

**Answer:**12Example-24

4 12 15 60 64 320 325 __.

Solution

**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 4
The second term is obtained by interchanging the digits of 5
So, we can get the missing term by interchanging the digits of 9
Hence, 18 is the missing number.

^{2}.^{2}and so on.^{2}.**Answer:**18