TITLE: Given a Fibonacci number , find the next Fibonacci number
QUESTION [49 upvotes]: The Fibonacci sequence is $0, 1, 1, 2, 3, 5, 8, 13, 21, 34,\ldots$, where each term after the first two is the sum of the two previous terms.
Can we find the next Fibonacci number if we are given any Fibonacci number?
For example, if $n = 8$ then the answer should be $13$ because $13$ is the next Fibonacci number after $8$.

REPLY [5 votes]: Along similar lines to Matthew Daly's answer:
Binet's formula gives an exact value for the n'th Fibonacci number where numbering starts with $F_0=0$ and $F_1=1$:
$F_n = \frac{(1+\sqrt{5})^n-(1-\sqrt{5})^n}{2^n \sqrt{5}}$ 
$\sqrt{5} F_n=(\frac{1+\sqrt{5}}{2})^n - (\frac{1-\sqrt{5}}{2})^n$
$=\phi^n - (\frac{-1}{\phi})^n$, where $\phi = \frac{1+\sqrt{5}}{2}$ (using the identity that $\phi-1=\frac{1}{\phi}$, which is easy to prove).
From there it's easy to show that for $n>2$, $|$log$_\phi(F_n\sqrt{5})-n|<0.5$. (Hint: as $n$ becomes large, the first of these two terms becomes very large and the second goes to zero.)
If $F_n=1$ obviously the question is unanswerable, and if $F_n=0$ it's trivial. If $F_n>1$ then $n>2$ and so we can calculate $n$ by rounding log$_\phi(F_n\sqrt{5})$ to the nearest integer. 
Now we have $n$, simply apply Binet's formula in the forwards direction and we're done.