Question 43
The real-valued function $f$ is defined for positive integers, and the positive integer $a$ satisfies
$$
\begin{gathered}
f(a)=f(1995), \quad f(a+1)=f(1996), \quad f(a+2)=f(1997) \\
f(n+a)=\frac{f(n)-1}{f(n)+1} \quad \text{ for all positive integers } n
\end{gathered}
$$
(i) Show that $f(n+4a)=f(n)$ for all positive integers $n$.
(ii) Determine the smallest possible $a$.
$$
\begin{gathered}
f(a)=f(1995), \quad f(a+1)=f(1996), \quad f(a+2)=f(1997) \\
f(n+a)=\frac{f(n)-1}{f(n)+1} \quad \text{ for all positive integers } n
\end{gathered}
$$
(i) Show that $f(n+4a)=f(n)$ for all positive integers $n$.
(ii) Determine the smallest possible $a$.