Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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?

share|cite|improve this question
up vote 5 down vote accepted

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

share|cite|improve this answer

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 $\det(M^n) = (\det(M))^n$.

share|cite|improve this answer

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

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.