\[\begin{array}{l} {\text{Пусть }}f\left( x \right) = \frac{{x + 1}}{{x + 2}},{\text{ }}f\left( {f\left( x \right)} \right) = \frac{{{a_2}x + {b_2}}}{{{b_2}x + {c_2}}},{\text{ }}f\left( {f\left( {f\left( x \right)} \right)} \right) = \frac{{{a_3}x + {b_3}}}{{{b_3}x + {c_3}}}{\text{ и т}}{\text{.д}}{\text{.}} \hfill \\ {a_n},{b_n},{c_n} \in \mathbb{N}.{\text{ Докажите}}{\text{, что }}{a_n}{c_n} = b_n^2 + 1. \hfill \\ \end{array} \]