What is the most elegant way to calculate this fraction?

Question

I can see some patterns, but still can’t find a way to calculate this fraction without doing it by hand. Just transcribing it is overwhelming enough.

Answer 1

Consider the sequence defined by

$a_0 = -1$
$a_{n + 1} = \frac{\frac{1}{5} + a_n}{1 - \frac{1}{5} a_n}$

Then we seek $\frac{1}{a_4}$ (it takes a bit of analysis to see this). We see that

$a_0 = -1$
$a_1 = -\frac{2}{3}$
$a_2 = -\frac{7}{17}$
$a_3 = -\frac{9}{46}$
$a_4 = \frac{1}{239}$

So the answer is $239$.