Question 8
(IND 1) Define a sequence $\langle f(n)\rangle_{n=1}^\infty$ of positive integers by $f(1)=1$ and
$$f(n)=\begin{cases}f(n-1)-n,&f(n-1)\gt n;\\f(n-1)+n,&f(n-1)\le n,\end{cases}$$
for $n\ge2$. Let $S=\{n\in\mathbb N\mid f(n)=1993\}$.
(a) Prove that $S$ is an infinite set.
(b) Find the least positive integer in $S$.
(c) If all the elements of $S$ are written in ascending order as $n_1\lt n_2\lt n_3\lt \cdots$, show that
$$\lim_{i\to\infty}\frac{n_{i+1}}{n_i}=3.$$
$$f(n)=\begin{cases}f(n-1)-n,&f(n-1)\gt n;\\f(n-1)+n,&f(n-1)\le n,\end{cases}$$
for $n\ge2$. Let $S=\{n\in\mathbb N\mid f(n)=1993\}$.
(a) Prove that $S$ is an infinite set.
(b) Find the least positive integer in $S$.
(c) If all the elements of $S$ are written in ascending order as $n_1\lt n_2\lt n_3\lt \cdots$, show that
$$\lim_{i\to\infty}\frac{n_{i+1}}{n_i}=3.$$