Prove the given property of the Fibonacci numbers

Question

I found in one of the books I read a lot of interesting properties of fibonacci numbers and among others this one in particular: For all $n \in \mathbb N$, $F_{n+1} F_{n-1} - F_n^2 = (-1)^n$.

I tried to use induction but at some point on induction step I just loose the point. Any ideas?

Answer 1

If $M = \pmatrix{0 & 1\cr 1& 1\cr}$, then $M^n = \pmatrix{F_{n-1} & F_n\cr F_n & F_{n+1}\cr}$ (easy to prove by induction). The left side of your equation is $\det(M^n) = (\det(M))^n$.

Answer 2

Induction will work fine. Assuming it works for $n$, we have $$F_{n+2}F_n-F_{n+1}^2=(F_{n+1}+F_n)F_n-F_{n+1}^2=F_{n+1}(F_n-F_{n+1})+F_n^2=F_{n+1}(-F_{n-1})+F_n^2$$ $$=-\left(F_{n+1}F_{n-1}-F_n^2\right)=-(-1)^n=(-1)^{n+1}$$

Answer 3

Hint: Use formula: $$F(n) = \frac{a^n - b^n}{\sqrt{5}}$$ With $a = \frac{1 + \sqrt{5}}{2}$, and $b=\frac{1 - \sqrt{5}}{2}$