Given the sequence $\left\{a_{\mathrm{n}}\right\}$ satisfies the recurrence relation $a_{\mathrm{a}}=8 a_{n-1}-6 a_{\mathrm{n}-2}$, and $a_{1}=1, a_{2}=5$, find its general term formula.
The sequence $\left\{a_{\mathrm{n}}\right\}$ satisfies a homogeneous linear recurrence relation with constant coefficients. Let $a_{\mathrm{n}}=A_{1} \cdot q_{1}^{\mathrm{n}}+A_{2} \cdot q_{2}^{\mathrm{n}}$, where $q_{1}, q_{2}$ are the roots of the characteristic equation $x^{2}-5 x+6=0$. Solving, we get $q_{1}=x_{1}=3, q_{2}=x_{2}=2$. Thus, $a_{\mathrm{n}}=A_{1} \cdot 3^{\mathrm{n}}+A_{2} \cdot 2^{\mathrm{n}}$.
Substituting $a_{1}=1, a_{2}=5$ into the equation, we get the system of equations
$$
\left\{\begin{array}{l}
3 A_{1}+2 A_{2}=1, \\
9 A_{1}+4 A_{2}=5 .
\end{array}\right.
$$

Solving, we get $A_{1}=1, A_{2}=-1$.
$$
\text { Hence } a_{n}=3^{n}-2^{n} \text {. }
$$