While doodling in class, I've discovered a peculiar recursive function:
$$a_0 = 1$$ $$a_1 = -2$$ $$a_{n+2}=\frac{a_{n+1}}{a_{n}},n\ge0$$ $$\begin{array}{c|c}n&a_n\\\hline0&1\\1&-2\\2&-2\\3&1\\4&-\frac12\\5&-\frac12\\6&1\\7&-2\\7&-2\\9&1\\10&-\frac12\\\vdots&\vdots\end{array}$$
It's a periodic function!! I haven't seen this in a recursion function.
Calculate the limit: $P(x,y)=x-\frac{x^2}{100y}$ (x - y%); $\mu_0 = 100$; $\mu_{n+1}=P(a_n,10)$. What is $\displaystyle\lim_{n\to\infty}\mu_n$, if any?
Answer: 0. Check it yourself.
