I want to check if $$\left\lfloor \left( \sum_{k=n}^{2n}{\frac{1}{F_{2k}}} \right)^{-1} \right\rfloor =F_{2n-1}~~(n\ge 3) \tag{$*$}$$ where $\lfloor x \rfloor$ is th floor function.

The Fibonacci sequence is defined by $F_1=1$, $F_2=1$, $F_{n+1}=F_n+F_{n-1}~(n\ge 2)$. Then we can get $$F_n=\dfrac{\alpha^n-\beta^n}{\sqrt{5}}$$ where $\alpha=\dfrac{1+\sqrt{5}}{2}$ and $\beta=\dfrac{1-\sqrt{5}}{2}.$

The following are some of my attempts:

For some example:

$n=3$, the left hand is $5$, the right hand is $5.$

$n=4$, the left hand is $13$, the right hand is $13.$

$$\vdots$$

$n=15$, the left hand is $514229$, the right hand is $514229.$

It is all true. But as $n$ increases, the order of magnitude grows very rapidly.

I ask one of my good friends to use a Python program to check $(*).$ He says it is true for $n\le 35$. When $n=36$, the Python says it is not true, But when $n= 37$, it is true again.

Thus I change one way and I ask my fiend to use a Python program to check $$\left( \sum_{k=n}^{2n}{\frac{1}{F_{2k}}} \right)^{-1} =F_{2n-1}~~(n\ge 3) \tag{$**$}.$$

Then the program shows it is true at least for $31\le n\le 51.$

But as you see, the left hand of $(**)$ is a decimal and the right hand of $(**)$ is an integer.

So I do not know if it is because the order of magnitude on the left hand of $(*)$ is growing very fast, $(*)$ becomes not true due to some computer shortcomings.

Finally I wonder if $(*)$ is true or false? Any help and references are greatly appreciated.

Thanks!

I have also posted it on MSE as On the finite sum of reciprocal Fibonacci sequences

J. Inequal. Appl.(2015), Article 73. doi.org/10.1186/s13660-015-0595-6 $\endgroup$