Need help finding the closed form of a sequence based upon the fibonacci sequence.

Question

I have been given an assignment question that asks for a simple closed form of the following sequence:

$$G_n=\left|\begin{array}{cc} F_n & F_{n+1}\\ F_{n+1} & F_{n+2} \end{array}\right|$$

I have tried taking the determinant, but substituting in the closed form of the Fibonacci sequence leads to nothing simple at all.

Thanks.

Answer 1

Using closed form of $F_n=\frac{\alpha^n-\beta^n}{\alpha-\beta}$ where $\alpha=\frac{-1+\sqrt{5}}{2},\ \beta=\frac{-1-\sqrt{5}}{2}$ will work, but maybe after a long and tedious calculation. A simpler way is to look at it in the following way.

$$G_n=F_{n}F_{n+2}-F_{n+1}^2=F_n(F_n+F_{n+1})-F_{n+1}^2\\=F_n^2-F_{n+1}(F_{n+1}-F_n)=F_n^2-F_{n+1}F_{n-1}=-G_{n-1}\\\implies G_n=(-1)^{n-1}G_1=(-1)^{n-1}$$