15.23. In how many ways can the number $n$ be represented as the sum of several addends, each equal to 1 or 2? (Representations differing in the order of the addends are considered different.)
15.23. Answer: $F_{n+1}$. Let the desired number be $x_{n}$. It is clear that $x_{1}=1$ and $x_{2}=2$. We will prove that $x_{n}=x_{n-1}+x_{n-2}$ for $n \geqslant 3$. Indeed, the number of representations of the number $n$ with the first summand 1 is $x_{n-1}$, and with the first summand $2-x_{n-2}$.