Definition:Value of Continued Fraction/Finite

Definition
Let $K$ be a field.

Let $n\geq0$ be a natural number.

Let $\left[{a_0, a_1, a_2, \ldots, a_n}\right]$ be a finite continued fraction in $K$.

Definition 1
Its value is recursively defined as:
 * $[a_0] = a_0$
 * $\left[{a_0, a_1, \ldots, a_n}\right] = a_0 + \dfrac 1 {\left[{a_1, \ldots, a_n}\right]}$

Definition 2
Let the matrix product:
 * $\begin{pmatrix}a_0 & 1 \\ 1 & 0\end{pmatrix}\cdots\begin{pmatrix}a_n & 1 \\ 1 & 0\end{pmatrix} = \begin{pmatrix}x_{11} & x_{12} \\ x_{21} & x_{22}\end{pmatrix}$

The value of the finite continued fraction is $\dfrac{x_{11}}{x_{21}}$