Side 2. Given the sequence $a_{n}=2 a_{n-1}-a_{n-2}$, where $a_{1}=2, a_{2}=3, n=3,4, \cdots$. Try to find the sum of the first $n$ terms and the general term formula of this sequence.
Given the recursive sequence, where $p=2$, $q=-1$, by Theorem 1 we have
$$
\begin{aligned}
S_{\mathrm{n}} & =n a_{1}+\frac{n(n-1)}{2}\left(a_{2}-a_{1}\right) \\
& =\frac{1}{2} n^{2}+\frac{3}{2} n, \\
a_{\mathrm{n}} & =S_{\mathrm{n}}-S_{\mathrm{n}-1} \\
& =\frac{1}{2} n^{2}+\frac{3}{2} n-\frac{1}{2}(n-1)^{2}-\frac{3}{2}(n-1) \\
& =n+1 . \\
& \left(S_{0}=0\right)
\end{aligned}
$$