Example 2 Let a sequence be
$2,12,36,80,150, \cdots$.
Determine the type of its general term formula and find the analytical expression.
Solve This problem is similar to Example 1. The corresponding data are as follows:
\begin{tabular}{ccccc}
$n$ & $P(n)$ & $\Delta P(n)$ & $\Delta^{2} P(n)$ & $\Delta^{3} P(n)$ \\
1 & 2 & & & \\
2 & 12 & 10 & 14 & \\
3 & 36 & 24 & & 6 \\
4 & 80 & 44 & 20 & 6
\end{tabular}
$5 \quad 150$
First, the third difference yields a constant 6, which indicates that $P(n)$ is a polynomial of degree 3.
Let $P(n)=a x^{3}+b x^{2}+c x+d$.
Then $a=\frac{6}{3!}=1$.
Next, subtract $x^{3}$ from the original expression (i.e., $P(n)-n^{3}$), to get new data, and repeat the process, ultimately obtaining
$$
P(x)=x^{3}+x^{2} .
$$