Example 1 (Bee Entering Hive Problem) A bee crawls from hive $A$ to the $n$-th hive, each time only moving to the adjacent hive on the right and never moving backward. How many different ways are there?
The number of routes for a bee to crawl to room 1 is $u_{1}=1$, to room 2 is $u_{2}=2$ routes $(A \rightarrow 2$ or $A \rightarrow 1 \rightarrow$ 2), $\cdots$
The routes for the bee to crawl to room $n$ fall into two categories:
One category is not passing through room $n-1$, but directly entering room $n$ from room $n-2$;
The other category is passing through room $n-1$.
The number of routes from $A$ to room $n-2$ is $u_{n-1}$, and the number of routes from $A$ to room $n-1$ is $u_{n}$, so the number of routes for the bee to crawl from $A$ to room $n$ is $u_{n-1}+u_{n}=u_{n+1}$. Clearly, $\left\{u_{n}\right\}$ is precisely the Fibonacci sequence (noting that $u_{1}=1$). Therefore,
$$
u_{n+1}=\frac{1}{\sqrt{5}}\left[\left(\frac{1+\sqrt{5}}{2}\right)^{n+1}-\left(\frac{1-\sqrt{5}}{2}\right)^{n+1}\right]
$$