Number Theory

Let's work from bottom up. Starting with the smallest, we have $\frac{1}{\frac{17}{4}}=\frac{4}{17}$

Then, we have $4+\frac{4}{17}=\frac{72}{17}$

$4+\frac{1}{4+\frac{1}{4}}=\frac{72}{17}$

Thus, we have

$\frac{1}{4+\frac{1}{4+\frac{1}{4}}}=4+\frac{17}{72}=\frac{305}{72}$

The final answer is $\frac{305}{72}$