Correct Option (3)

The given numerical series is 7, X, 21, 31, 43. To determine the missing number 'X', the pattern of differences between consecutive terms is analyzed.

The difference between the fourth term (31) and the third term (21) is 10.
The difference between the fifth term (43) and the fourth term (31) is 12.

This establishes an arithmetic progression of differences, where each subsequent difference increases by 2. Following this pattern, the differences between terms should be 6, 8, 10, 12.

Applying this to the earlier terms: The difference between the third term (21) and the missing term (X) should be 8. Thus, X = 21 - 8 = 13.
Alternatively, the difference between the missing term (X) and the first term (7) should be 6. Thus, X = 7 + 6 = 13.

Both calculations consistently identify the missing number X as 13.

Incorrect Options:

If X were 11 (Option 1), the differences would be (11 - 7) = 4 and (21 - 11) = 10. This sequence of differences (4, 10) does not align with the established pattern of increasing differences by 2 (6, 8, 10, 12).
If X were 12 (Option 2), the differences would be (12 - 7) = 5 and (21 - 12) = 9. This sequence of differences (5, 9) does not conform to the established pattern.
If X were 14 (Option 4), the differences would be (14 - 7) = 7 and (21 - 14) = 7. This sequence of differences (7, 7) does not conform to the established pattern of increasing differences.