16. Suppose you have 100 RMB to spend on items, with the rule that you can only buy one of the following three items each day: (1) candy for 1 RMB; (2) cookies for 2 RMB; (3) fruit for 2 RMB, and you are not allowed to buy nothing. How many ways are there to spend all 100 RMB?
16. First, consider the number of ways to spend $n$ yuan as $x_{n}$. Categorize the spending method for the first day: If 1 yuan is spent on a snack on the first day, the remaining $(n-1)$ yuan can be spent in the following days, and the number of ways is $x_{n-1}$.

If 2 yuan is spent on cookies or fruit on the first day, the remaining $(n-2)$ yuan can be spent in the following days, and the number of ways is $x_{n-2}$. Therefore, we have $x_{n}=x_{n-1}+2 x_{n-2} \ (n \geqslant 3)$, and $x_{1}=1, x_{2}=3$. Thus, we can derive $x_{n}=2^{n}-x_{n-1}=\cdots=\frac{1}{3}\left[2^{n+1}+(-1)^{n}\right]$. Substituting $n=100$ gives $x_{100}=\frac{1}{3}\left(2^{101}+1\right)$, meaning the number of ways to spend 100 yuan is $\frac{1}{3}\left(2^{101}+1\right)$.