MathDB

Define a sequence

\langle f(n)\rangle^{\infty}_{n=1}

of positive integers by

f(1) = 1

and

f(n) = \begin{cases} f(n-1) - n & \text{ if } f(n-1) > n;\\ f(n-1) + n & \text{ if } f(n-1) \leq n, \end{cases}

for

n \geq 2.

Let

S = \{n \in \mathbb{N} \;\mid\; 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.

Problem Statement