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 $F_n^{-1}$)?
Best Answer
For part (3), $F_1=F_2=1$ so you cannot hope for an inversion formula which works for all $n$. For large $n$, however, the term in $\phi^{-n}$ becomes very small and $F_n$ is the nearest integer to $\frac {\phi^n} {\sqrt 5}$ and it is very nearly true that$$n=\frac{\log {(F_n\sqrt 5)}}{\log {\phi}}$$