= Solution
Induct on $n$, with the claim quantified over every $m\ge n$. At $n=0$,
$$
F_mF_1-F_0F_{m+1}=F_m=(-1)^0F_m.
$$
Suppose the identity holds for a fixed $n$ and every $m\ge n$. For $m\ge n+1$, the <Fibonacci recurrence> gives
$$
\begin{aligned}
F_mF_{n+2}-F_{n+1}F_{m+1}
&=F_m(F_{n+1}+F_n)-F_{n+1}(F_m+F_{m-1})\\
&=-\big(F_{m-1}F_{n+1}-F_nF_m\big)\\
&=-(-1)^nF_{m-1-n}=(-1)^{n+1}F_{m-(n+1)}.
\end{aligned}
$$
Applying the <induction hypothesis> with $m-1\ge n$ is legitimate. This proves \b[$F_mF_{n+1}-F_nF_{m+1}=(-1)^nF_{m-n}$] throughout its domain, a case of the <Fibonacci determinant identity>.
Back to article page
