7. [5] An infinite sequence of positive real numbers is defined by $a_{0}=1$ and $a_{n+2}=6 a_{n}-a_{n+1}$ for $n=0,1,2, \cdots$ Find the possible value(s) of $a_{2007}$.
Answer: $\mathbf{2}^{\mathbf{2 0 0 7}}$. The characteristic equation of the linear homogeneous equation is $m^{2}+m-6=$ $(m+3)(m-2)=0$ with solutions $m=-3$ and $m=2$. Hence the general solution is given by $a_{n}=A(2)^{n}+B(-3)^{n}$ where $A$ and $B$ are constants to be determined. Then we have $a_{n}>0$ for $n \geq 0$, so necessarily $B=0$, and $a_{0}=1 \Rightarrow A=1$. Therefore, the unique solution to the recurrence is $a_{n}=2^{n}$ for all $n$.