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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.

This is the fraction.

JMP
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    The way to translate the pattern you are seeing is write it down as a recurrence. – dxiv May 25 '21 at 05:10
  • Are you sure the bottom level is $\frac{\frac{1}{5}-1}{\frac{1}{5}+1}$ and not $\frac{1-\frac{1}{5}}{\frac{1}{5}+1}$? – 1Rock May 25 '21 at 05:13
  • @1Rock I kind of see why you are asking this, but no, this is how I found the fraction. –  May 25 '21 at 05:44

1 Answers1

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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 $239$.

Mark Saving
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