The First Continued Fraction Identity
\[[a_1,a_2,...,a_n ] =a_1+\frac{1}{a_2+\frac{1}{a_3+ \frac{1}{\ddots a_{n-1}+1/a_n}}}\]
. Example:
\[\begin{equation} \begin{aligned} \frac{225}{157} &= 1+\frac{68}{157}=1+\frac{1}{157/68} \\ &=1+\frac{1}{2+21/68}=1+\frac{1}{2+\frac{1}{68/21}} \\ &= 1+\frac{1}{2+\frac{1}{3+ \frac{5}{21}}} = 1+\frac{1}{2+\frac{1}{3+\frac{1}{21/5}}} \\ &= 1+\frac{1}{2+\frac{1}{3+\frac{1}{4+\frac{1}{5}}}}= [ 1,2,3,4,5 ] \end{aligned} \end{equation}\]
The identity holds also for infinite continued fractions. 
