[Continued (infinite) fractions]

Calculate the following continued fractions:
a) $[5 ;(1,2,1,10\}]$
b) $[5 ;(1,4,1,10\}] ;$
c) $[2 ;(1,1,3\}]$.
a) Consider the number $\alpha=[10 ;(1,2,1,10)]$. Then $\alpha=10+\frac{1}{1+\frac{2}{2+\frac{1}{1+\frac{1}{\alpha}}}}=\frac{43 \alpha+32}{4 \alpha+3}$. From this, $4 \alpha^{2}-40 \alpha-$ $32=0$, that is, $\alpha^{2}-10 \alpha-8=0$. Since $\alpha>0$, we get $\alpha=5+\sqrt{33}$. The desired number is $\alpha-5$.

$$
\text { a) } \sqrt{33} \text {; b) } \sqrt{34} \text {; c) } \frac{1+\sqrt{17}}{2} \text {; }
$$

Problem