I'm interested in solving the equation $$ \color{red}{x}=1+\cfrac{a_1}{b_1+\cfrac{a_2}{b_2+\ddots \cfrac{a_n}{b_n+\color{red}{x}}}}, $$ where $a_i,b_i$ are positive real numbers. Is there a formula to simplify this continued fraction?

Any help will be appreciated.