Example 2 In the sequence $\left\{a_{n}\right\}$, $a_{1}=1, a_{2}=2, a_{n+2}=5 a_{n+1}-6 a_{n}, n=1,2$, $3, \cdots$, find the general formula for $\left\{a_{a}\right\}$.
The characteristic equation of the recurrence relation is $t^{2}-5 t+6=0$, with solutions $a=2, \beta=3$. Therefore, the general term formula for the sequence $\left\{a_{n}\right\}$ is $a_{n}=A \cdot 2^{n-1}+B \cdot 3^{n-1}$. Substituting $a_{1}=1, a_{2}=2$ into the above formula yields $\left\{\begin{array}{l}A+B=1 \\ 2 A+3 B=2,\end{array}\right.$ Solving the system of equations gives $A=1, B=0$. Thus, the general term formula for the sequence $\left\{a_{n}\right\}$ is $a_{n}=2^{n-1}$.