Školjka

%V0 Define a sequence $<f(n)>^{\infty}_{n=1}$ of positive integers by $$f(1) = 1$$ and $f(n) =\left\{\begin{array}{ll}f(n-1)-n &\text{ if }f(n-1) > n;\\ f(n-1)+n &\text{ if }f(n-1)\leq n,\end{array}\right$ for $n \geq 2.$ Let $S = \{n \in \mathbb{N} | f(n) = 1993\}.$ (i) Prove that $S$ is an infinite set. (ii) Find the least positive integer in $S.$ (iii) If all the elements of $S$ are written in ascending order as $$n_1 < n_2 < n_3 < \ldots ,$$ show that $$\lim_{i\rightarrow\infty} \frac{n_{i+1}}{n_i} = 3.$$