Find $x$ as the given $n$th term in the Fibonacci sequence?

Question

With a given $n$ and I am trying to find the value of $x$, as in: $$Fib(x)=n$$

Using the formula for Fibonacci sequence, where $\varphi$ is the Golden Ration ($\approx1.61803399\ldots$) $$Fib(z) = \frac{\varphi^z - \cos(z \pi)\varphi^{-z}}{\sqrt{5}}$$ How can I find $x$ if, lets say, $n=11261957$? I so far have this: $$\varphi^x-\cos(x\pi)\varphi^{-x}=\sqrt{5}(11261957)$$

But I don't know where to go from there. Any help is appreciated!

Answer 1

From $$\phi^x - \cos(x\pi)\phi^{-x} = 11261957\sqrt{5}$$ we can assume that $\phi^{-x}$ is small and get $$\phi^x \approx 11261957\sqrt{5}$$ The solution to this is $$x \approx \frac{ln(11261957\sqrt{5})}{ln(\phi)} \approx 35.414$$ and indeed the $35^\text{th}$ and $36^\text{th}$ Fibonacci numbers are $$9227465 \qquad \text{and} \qquad 14930352$$ which are as close as you can get to $11261957$.

Answer 2

You can calculate it recursively given the formula $$F_n=F_{n-2}+F_{n-1}$$(you can use a code which does it). Formulas which are iterative like $$(\omega=\frac{1+\sqrt 5}{2})\quad F_n=\frac{\omega^{n+1}-\bar\omega^{n+1}}{\sqrt 5}$$ are not that useful in your case, since that's equivalent to solve an equation which for big values of $F_n$ is not that easy to solve. Nevertheless, you can solve them using numerical methods taking $f(n)$ as the nth Fibonacci number.