Example 1. Given the sequence $\left\{a_{\mathrm{n}}\right\}$, where $a_{1}=1$. It satisfies the relation $a_{\mathrm{n}}=a_{\mathrm{n}-1}+2 n(n \geqslant 2, n \in N)$, find $a_{n}$.
Solve: Since $a_{n}-a_{n-1}=2n$, and the sequence with the general term $2n$ is an arithmetic sequence, the sequence $\left\{a_{\mathrm{n}}\right\}$ is a second-order arithmetic sequence, whose general formula is $a_{n}=A n^{2}+B n+C$, and it is only necessary to determine the coefficients $A, B, C$.

Given, we have $a_{1}=1, a_{2}=5, a_{3}=11$, substituting into the general formula, we get the system of equations.
$$
\left\{\begin{array}{l}
A+B+C=1 \\
4 A+2 B+C=5 \\
9 A+3 B+C=11
\end{array}\right.
$$

Solving, we get $A=1, B=1, C=-1$. Therefore, $a_{\mathrm{n}}=n^{2}+n-1$.