Given the Fibonacci sequence $F_n$, Wikipedia says (http://en.wikipedia.org/wiki/Fibonacci_number#List_of_Fibonacci_numbers) $$ F_{2n-1} = F_n^2+F^2 _{n-1}$$ so that $$F_{2n-1}>F^2_n$$
What is the smallest such k for which $$F_{n+k}>F^2_n\,\,?$$
I'm not sure where to start or to find smaller values than $2n-1$
$k=n-1$ is the minimum possible value.
We have $F_{2n-1}=F_n^2+F_{n-1}^2$. Rewrite the LHS as $F_{2n-2}+F_{2n-3}$, so $$F_{2n-2}=F_n^2+F_{n-1}^2-F_{2n-3}$$ But now $F_{2n-3}=F_{n-1}^2+F_{n-2}^2$. Substituting, we get $$F_{2n-2}=F_n^2-F_{n-2}^2<F_n^2$$ Hence even the term immediately preceding $F_{2n-1}$ is less than $F_n^2$.