$\begin{array}{l}A = 2+\dfrac{1}{1 + \dfrac{1}{2+ \dfrac{1}{2}}}\\ \to A = 2+\dfrac{1}{1 + \dfrac{1}{\dfrac{5}{2}}}\\ \to A = 2+\dfrac{1}{1 + \dfrac{2}{5}}\\ \to A = 2+\dfrac{1}{\dfrac{7}{5}}\\ \to A = 2 + \dfrac{5}{7}\\ \to A = \dfrac{19}{7} \end{array}$