Question

1 2 13
7 14 91
9 ? 117
Find the answer with logic ​

Answer 1

Number series is important for various competitive examinations. In this category of questions, a series of various numbers is given with a blank . We are supposed to find out a pattern between every number and its predecessor and find out the answer using the same logic.

For the purpose of better understanding of the concept, we classify them into the following categories:

Series with a constant difference

Series with an increasing difference

Series with a decreasing difference

Squares/ Cubes series

Combination of different operations

Miscellaneous

The best approach :

The best way of approaching number series questions is to observe the difference between various terms. If we see a constant difference, then it’s a constant difference series. If the difference is decreasing or increasing by a constant number , then it is a series of type 2 or type 3. But if there is no such increase or decrease in the difference, then try dividing the 2nd term with the first, third with the second and so on. If you obtain the same number each time, then it is a product series.

Also , if none of these seem to work, then you can  try writing each term as a product of two factors and try to see if there is any pattern. If you still observe no pattern and the difference is increasing or decreasing rapidly, then try to look for square/ cube series.

If the difference is increasing and decreasing in some fixed manner, then it is a type of combination series.

Let us look at each type of series in greater depth:  

1. Series with a constant difference

In this kind of series, any 2 consecutive numbers have the same difference between them.

For example : 1 , 5 , 9 , 13 , ?

We can observe that we are adding 4 to the previous number to obtain the next number. So, answer here will be 13+4 = 17.

2. Series with an increasing difference

In this type of series, the difference between two consecutive terms keep on increasing as we move forward in a series. Let us try to use this theory in a question.

1,2,4,7,11,16,?

We can clearly observe that the series is increasing with the difference : +1, +2, +3 ,+4 , +5.

So, we will obtain our number by adding 6 to 16 which gives us 22.

3. Series with a decreasing difference

In this type of series, the difference between two consecutive terms keep on decreasing as we move forward in a series. Let us try to use this with some modification in the previous question that we did.

16,11,7,4,2, ?

We can clearly observe that the series is decreasing with the difference : -5, -4, -3 ,-2 .

So, we will obtain our number by subtracting 1 from 2 which gives us 1.

4. Squares/ Cubes series

We can have series where the terms are related to the squares/ cubes of numbers. We can have a lot of variations here. Let us look at some of the possibilities.

1, 9, 25, 49 , ?

We can observe that the above series is square of odd numbers starting from one. So our answer will be 9^2 = 81.

Let us look at another example:

1 , 1 , 2 , 4 , 3 , 9 , 4 , ?

We observe here that the series is formed by writing numbers starting from 1 along with its square as the next number i.e. ( 1 , 12) , (2, 22) and so on. So we obtain our answer as 16 which is 42.

Consider the following question:

9 , 28 , ? , 126.

The answer for above question will be 65, let us discuss how.

9         ,                 28   ,     ?     ,    126.

( 23+1)   (33+1)                 (53+1)

The blank should have 43+1. Hence, the answer is 65.

5. Combination of different operations

This kind of series has more than 1 type of arithmetic operations which have been performed or it can also have 2 different series which have been combined to form a single series. This kind of series is the the most asked and the most important among all the types of series that we have discussed so far.

Consider the series:

1, 3 , 6 , 2 , 6 , 9 , 3 , 9 , ?

The first term 1 is multiplied by 3 to give the second term, 3 has been added to the second term to get the third term. The next term is 2 which is 1 more than the 1st term. It is multiplied with 3 to give next term and the process is continued. With this process, we obtain our answer as 12.

Consider the series:

6, 10 , 7, 11 , 8 , 12 , ?

We can see that the above series is a combination of 2 simple series:

1st , 3rd , 5th terms make an increasing series of 6 , 7 , 8….. . The 2nd , 4th and 7th term make a series of 10 , 11 , 12… . So, our answer will be 9 which is the 7th term of the original series.