Question 31
Suppose that a function $f$ defined on the positive integers satisfies
$$
\begin{gathered}
f(1)=1, \quad f(2)=2 \\
f(n+2)=f(n+2-f(n+1))+f(n+1-f(n)) \quad(n \geq 1)
\end{gathered}
$$
a) Show that
(i) $0 \leq f(n+1)-f(n) \leq 1$
(ii) if $f(n)$ is odd, then $f(n+1)=f(n)+1$.
b) Determine, with justification, all values of $n$ for which
$$
f(n)=2^{10}+1
$$
$$
\begin{gathered}
f(1)=1, \quad f(2)=2 \\
f(n+2)=f(n+2-f(n+1))+f(n+1-f(n)) \quad(n \geq 1)
\end{gathered}
$$
a) Show that
(i) $0 \leq f(n+1)-f(n) \leq 1$
(ii) if $f(n)$ is odd, then $f(n+1)=f(n)+1$.
b) Determine, with justification, all values of $n$ for which
$$
f(n)=2^{10}+1
$$