Correct Option (2)

The given sequence is 2, 12, 36, 80, 150, X.

To determine the value of X, we analyze the differences between consecutive terms:

First differences:
12 - 2 = 10
36 - 12 = 24
80 - 36 = 44
150 - 80 = 70

The sequence of first differences is 10, 24, 44, 70.

Next, we analyze the differences between consecutive terms of the first differences (second differences):

Second differences:
24 - 10 = 14
44 - 24 = 20
70 - 44 = 26

The sequence of second differences is 14, 20, 26.

Observing the second differences, we find a consistent pattern: each term increases by 6 (20 - 14 = 6, 26 - 20 = 6). This indicates an arithmetic progression in the second differences with a common difference of 6.

To find the next term in the second differences: 26 + 6 = 32.

To find the next term in the first differences: 70 + 32 = 102.

Finally, to find the value of X: X = 150 + 102 = 252.

Incorrect Options:

Options 1 (248), 3 (258), and 4 (262) are incorrect because they do not conform to the established mathematical pattern derived from the sequence's first and second differences. The unique arithmetic progression identified in the second differences leads specifically to 252 as the next term in the sequence.