Example 2. The sequence $\left\{\mathrm{a}_{\mathrm{n}}\right\}$ satisfies the conditions $\mathrm{a}_{1}=1$,
$$
\begin{array}{c}
\mathrm{a}_{2}=-\frac{3}{2}, 4 \mathrm{a}_{\mathrm{k}+2}-12 \mathrm{a}_{\mathrm{k}+1}+9 \mathrm{a}_{\mathrm{k}}=0 . \\
(\mathrm{k}=1,2,3, \cdots) \text { Find the general term } \mathrm{a}_{\mathrm{n}} .
\end{array}
$$
Solving $4 x^{2}-12 x+9=0$, we get $x_{1}=x_{2}=\frac{3}{2}$. By Theorem 2, let $a_{n}=\left(A_{0}+n B_{0}\right)$
- $\left(\frac{3}{2}\right)^{n-1}$, from $a_{1}=1, a_{2}=-\frac{1}{2}$ we get the system of equations
$$
\left\{\begin{array}{l}
\mathrm{A}_{0}+\mathrm{B}_{0}=1, \\
\frac{3}{2} \mathrm{~A}_{0}+3 \mathrm{~B}_{0}=-\frac{3}{2} . \\
\mathrm{A}_{0}=3, \mathrm{~B}_{0}=-2 .
\end{array}\right.
$$

Therefore, the general term of the sequence is
$$
a_{n}=(3-2 n)\binom{3}{2}^{n-1} \text {. }
$$