Correct Option (C)

The given sequence is 14, 18, 20, 24, 30, 32. An analysis of the numbers reveals a pattern where each term is one greater than a consecutive prime number:

14 = 13 + 1 (13 is a prime number)
18 = 17 + 1 (17 is a prime number)
20 = 19 + 1 (19 is a prime number)
24 = 23 + 1 (23 is a prime number)
30 = 29 + 1 (29 is a prime number)
32 = 31 + 1 (31 is a prime number)

Following this established pattern, the next prime number after 31 is 37. Therefore, the next number in the sequence would be 37 + 1 = 38.

Incorrect Options:

Option A (34): If 34 were the next number, it would imply 34 = 33 + 1. However, 33 is not a prime number (33 = 3 × 11), which violates the established pattern.
Option B (36): If 36 were the next number, it would imply 36 = 35 + 1. However, 35 is not a prime number (35 = 5 × 7), which violates the established pattern.
Option D (40): If 40 were the next number, it would imply 40 = 39 + 1. However, 39 is not a prime number (39 = 3 × 13), which violates the established pattern.