What is the missing number ‘X’ of the series

7, X , 21, 31, 43 ?

The correct option is 13.

Let us analyze the pattern of the given series to determine the value of the missing number ‘X’:
The series is: 7, X, 21, 31, 43

To find the underlying pattern, let us examine the differences between the consecutive known terms at the end of the series:

The difference between the fourth term and the third term is:
31 - 21 = 10

The difference between the fifth term and the fourth term is:
43 - 31 = 12

Notice that the differences between the consecutive numbers are consecutive even numbers (10 and 12). If we extend this pattern backwards, the differences should be consecutive even numbers increasing by 2 at each step. Thus, the sequence of differences between consecutive terms should be:
6, 8, 10, 12

Let us apply this pattern to find the missing term ‘X’:
The difference between the second term (X) and the first term (7) should be 6:
X - 7 = 6
X = 7 + 6 = 13

Let us verify if this value of X satisfies the next difference in the pattern, which should be 8:
The difference between the third term (21) and the second term (X = 13) is:
21 - 13 = 8

Since the differences (6, 8, 10, 12) form a consistent arithmetic progression of even numbers, the value of the missing term is confirmed to be 13.