<s>[INST] Finding $a + b + c$ given that $\;a + \frac{1}{b+\large\frac 1c} = \frac{37}{16}$ Please help me to find the needed sum:

If $a,b,c$ are positive integers such that $\;a + \dfrac{1}{b+\large \frac 1c} = \dfrac{37}{16},\;$
  find the value of $\;(a+b+c)$.

Thanks!
 [/INST] This is asking for the Continued Fraction expansion of $\frac{37}{16}$. We can use the Euclid-Wallis Algorithm to compute the expansion:
$$
\begin{array}{r}
&&2&3&5\\\hline
1&0&1&-3&16\\
0&1&-2&7&-37\\
37&16&5&1&0\\
\end{array}
$$
The continued fraction is above the horizontal line:
$$
\frac{37}{16}=2+\cfrac1{3+\cfrac1{5}}
$$
Then, $a+b+c=2+3+5=10$.
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