Question

Suppose that \(n\) is a positive integer. Show that we can write
\[
n=F_{i_{1}}+F_{i_{2}}+\cdots+F_{i_{k}}
\]
where \(k\) is a positive integer, \(1 \leq i_{1}\) and \(i_{j} \geq i_{j-1}+2\) for \(j=2,3, \ldots, k\). Also show that \(k\) and \(i_{1}, \ldots, i_{k}\) are uniquely determined by \(n\).