Question

Find the missing number in the series $ 2,10,26,.....,242 $
A. $ 80 $
B. $ 81 $
C. $ 82 $
D. $ 84 $

Answer 1

Hint: In order to find the missing number of the series, we must know what kind of pattern is being followed by the series, whether the series followed by an AP, GP or any other pattern is there. In these problems we can only assume the pattern that is being followed, checking whether all the terms follow the same pattern or not.

Complete step-by-step answer:
We are given a series $ 2,10,26,.....,242 $ .
We need to find the missing value from the series. Through the series, we can see that the pattern is neither AP (Arithmetic Progression) nor GP (Geometric Progression), so we move on checking what other pattern is being followed.
For, the first term we assume it to be written as:
 $ 2 = {3^1} - 1 $ ……..(1)
Similarly, if we check for the second term, we can write it as:
 $ 10 = {3^{1 + 1}} + 1 $
 $ \Rightarrow 10 = {3^2} + 1 $ ……(2)
Now, if it follows the same pattern in the whole series, then the third term should follow the same rules as equation 1. And, we check it as:
 $ 26 = {3^{2 + 1}} - 1 $
 $ \Rightarrow 26 = {3^3} - 1 $
And, we can see that this pattern is being followed, the power of three is increasing by each term and in odd terms one is subtracted and in even terms one is being added.
Similarly, we can get the missing term following the same method as equation 2, and we get:
Missing terms as:
 $ {3^{3 + 1}} + 1 $
 $ \Rightarrow {3^4} + 1 $
 $ \Rightarrow 81 + 1 = 82 $
And, the missing number we get is $ 82 $ , but we again follow the pattern in order to get and check the next term:
 $ {3^{4 + 1}} - 1 $
 $ \Rightarrow {3^5} - 1 $
 $ \Rightarrow 243 - 1 = 242 $
Which is the exact next term. So, the missing number is correct.
Hence, Option C – 82 is correct.
So, the correct answer is “Option C”.

Note: There is no specific formula that is being followed. We need to check it yourself and assume and develop a pattern that perfectly matches with the series.
We can cross check our answers for a missing pattern by checking the previous or next term of that missing number using the number found. If it satisfies then it’s the correct answer.