TITLE: Fibonacci nth term
QUESTION [5 upvotes]: It is known that the nth term of the Fibonacci sequence can be found by the formula:
$F_n = \frac{\phi^n - (-\phi)^{-n}}{\sqrt{5}}$,
where $\phi$ is the golden ratio (1.618...).

Would this be the best formula to generate large terms of the sequence (eg. $n = 10^{15}$)?
How many decimal places of $\phi$ should be known to generate such a large term?
How can this formula be reversed (ie. finding $F_n^{-1}$)?

REPLY [2 votes]: For the accuracy of $\phi$,  as $F_n \approx \frac {\phi^n}{\sqrt 5}$, the fractional error in $F_n$ is the same as your fractional error in $\phi^n$.  If instead of $\phi^n$ you have $(\phi(1+\epsilon))^n$ the fractional error is $(1+\epsilon)^n \approx 1+n\epsilon$.  If you want $F_n$ to be exact, you need $\frac {n\epsilon}{\phi^n}\lt 1$ or $\epsilon \lt \frac {\phi^n}n$