What is the value of X in the sequence 2, 12, 36, 80, 150, X?

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

1. Calculate the first differences between consecutive terms: 12 - 2 = 10 36 - 12 = 24 80 - 36 = 44 150 - 80 = 70 The new sequence of first differences is: 10, 24, 44, 70.

2. Calculate the second differences from the first differences: 24 - 10 = 14 44 - 24 = 20 70 - 44 = 26 The new sequence of second differences is: 14, 20, 26.

3. Calculate the third differences from the second differences: 20 - 14 = 6 26 - 20 = 6 The third differences are constant: 6, 6.

4. Extend the pattern to find X: Since the third difference is constant at 6, the next third difference will also be 6. The next second difference will be 26 + 6 = 32. The next first difference will be 70 + 32 = 102. The next term in the original sequence (X) will be the last term (150) plus the next first difference (102). X = 150 + 102 = 252.

Alternatively, the pattern can be expressed as n^2 * (n+1) for the nth term: For n=1: 1^2 * (1+1) = 1 * 2 = 2 For n=2: 2^2 * (2+1) = 4 * 3 = 12 For n=3: 3^2 * (3+1) = 9 * 4 = 36 For n=4: 4^2 * (4+1) = 16 * 5 = 80 For n=5: 5^2 * (5+1) = 25 * 6 = 150 For n=6 (X): 6^2 * (6+1) = 36 * 7 = 252.

The value of X is 252.

The final answer is B.