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

The correct option is 252.

To find the value of X in the sequence 2, 12, 36, 80, 150, X, we can analyze the mathematical pattern connecting the terms. Let us look at two different ways to solve this problem.

Method 1: Formula Pattern ($n^3+n^2$)
If we represent each term position as $n$ (where $n=1,2,3,...$), we can check if there is a polynomial relationship. Calculating the sum of the cube and the square of each position index yields the terms of the sequence:

For $n=1$:
$1^3+1^2=1+1=2$

For $n=2$:
$2^3+2^2=8+4=12$

For $n=3$:
$3^3+3^2=27+9=36$

For $n=4$:
$4^3+4^2=64+16=80$

For $n=5$:
$5^3+5^2=125+25=150$

Following this established pattern, the next term ($n=6$) is calculated as:

$X=6^3+6^2=216+36=252$

Method 2: Difference Analysis
We can analyze the differences between consecutive terms to find the underlying progression:

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

Second-level differences (differences of the first-level differences):
24 - 10 = 14
44 - 24 = 20
70 - 44 = 26

Third-level differences:
20 - 14 = 6
26 - 20 = 6

Since the third-level differences are constant at 6, we can work our way back up to find the next value in the sequence:
Next second-level difference = 26 + 6 = 32
Next first-level difference = 70 + 32 = 102
Next term in the sequence (X) = 150 + 102 = 252