15.22. What is the number of subsets of the set $\{1,2,3, \ldots, n\}$ that do not contain two consecutive numbers?
15.22. Answer: $F_{n+2}$. Let the desired number be $x_{n}$. For $n=1$ there are subsets $\varnothing$ and $\{1\}$, so $x_{1}=2$. For $n=2$ there are subsets $\varnothing, \{1\}$, and $\{2\}$, so $x_{2}=3$. It is easy to verify that $x_{n}=x_{n-1}+x_{n-2}$ for $n \geqslant 3$. Indeed, if the considered subset contains $n$, then it does not contain $n-1$, and no other restrictions apply to it. The number of such
subsets is $x_{n-2}$. It is also clear that the number of considered subsets that do not contain $n$ is $x_{n-1}$.