Example 1. If the sequence $\left\{a_{n}\right\}$ satisfies $a_{n+1}=a_{n}+3n+2, a_{1}=2$, find its general term formula.
$$
\begin{array}{c}
\text { Sol } \because a_{n}-a_{1}=\sum_{\mathrm{k}=1}^{\mathrm{n}-1}\left(a_{k+1}-a_{k}\right) \\
=\sum_{\mathrm{k}=1}^{\mathrm{n}-1}(3 k+2)=\frac{3 n^{2}+n-4}{2}, \\
\therefore a_{n}=\frac{1}{2}\left(3 n^{2}+n\right) .
\end{array}
$$

If $\sum_{\mathrm{k}=1}^{\mathrm{n}-1} f(k)$ can be found, then the general term formula of the sequence satisfying the recurrence relation $a_{n+1} - a_{n} = f(n), a_{1} = a$ can also be derived in this way.