I am wondering how to identify a Fibonacci number using Binet's formula. One of the approaches I tried was:
Using Binet's formula, $$F_n=\frac{\varphi^{n}-(-1)^n\varphi^{-n}}{\sqrt{5}}$$
I multiply the equation by $\sqrt{5}$, and by solving the quadratic, I get $$\varphi^n=\frac{F_n\sqrt{5}\pm\sqrt{5F_n+4(-1)^n}}{2}$$
I have done this by using wikipedia as my reference, but I don’t know how to get this:
$$\varphi^n=F_n\varphi+F_{n-1}=\frac{(F_n\sqrt{5}+F_n+2F_{n-1})}{2}$$
I have tried doing the following:
Using Binet’s formula,
$$\varphi^n=\frac{F_n\sqrt{5}\pm\sqrt{5F_n^2+4(-1)^n}}{2}=\frac{F_n5\sqrt{5}\pm5\sqrt{5F_n^2+4(-1)^n}}{10}$$
Using the formula:$\left(\frac{\varphi^n-(-1)^n\varphi^{-n}}{\sqrt{5}}\right) $,
$$\varphi^n=F_n\varphi+F_{n-1}$$
$$=\left(\frac{\varphi^n-(-1)^n\varphi^{1-n}}{\sqrt{5}}\right)\varphi+\left(\frac{\varphi^{n-1}-(-1)^n\varphi^{1-n}}{\sqrt{5}}\right)$$
$$=\frac{\varphi^{n+1}-2[(-1)^n\varphi^{1-n}]-\varphi^{n-1}}{\sqrt{5}}$$
$$=\frac{\varphi(F_{n+1}-F_{n-1})-2[(-1)^n\varphi^{1-n}]}{\sqrt{5}}$$
This boxed part in the image is the part I’m confused with.
Can you please explain to me how to get the boxed part? Thanks!
Best Answer
Once you have $\varphi^n = F_n \varphi + F_{n-1}$, just use $\varphi = (\sqrt{5}+1)/2$ to get $$ \varphi^n = F_n \frac{\sqrt{5}+1}{2} + F_{n-1} = \frac{F_n \sqrt{5} + F_n + 2 F_{n-1}}{2}$$