To determine the next number in the series [tex]\(2, 12, 36, 80, 150, ?\)[/tex], we start by examining the differences between consecutive terms.
1. Calculate the first differences:
[tex]\[
\begin{align*}
12 - 2 &= 10 \\
36 - 12 &= 24 \\
80 - 36 &= 44 \\
150 - 80 &= 70
\end{align*}
\][/tex]
So the first differences are [tex]\(10, 24, 44, 70\)[/tex].
2. Calculate the second differences (differences of the first differences):
[tex]\[
\begin{align*}
24 - 10 &= 14 \\
44 - 24 &= 20 \\
70 - 44 &= 26
\end{align*}
\][/tex]
So the second differences are [tex]\(14, 20, 26\)[/tex].
3. Calculate the third differences (differences of the second differences):
[tex]\[
20 - 14 = 6 \\
26 - 20 = 6
\][/tex]
The third differences are [tex]\(6, 6\)[/tex], indicating that the third difference is constant.
Since the third difference is constant, we can predict the next second difference by adding [tex]\(6\)[/tex] to the last second difference:
[tex]\[
26 + 6 = 32
\][/tex]
Next, we find the next first difference by adding the new second difference [tex]\(32\)[/tex] to the last first difference:
[tex]\[
70 + 32 = 102
\][/tex]
Finally, we find the next term in the series by adding this new first difference [tex]\(102\)[/tex] to the last term [tex]\(150\)[/tex]:
[tex]\[
150 + 102 = 252
\][/tex]
Therefore, the next number in the series is:
[tex]\[
\boxed{252}
\][/tex]
