SE-SKOLORNAS 1965 Problem 4

Find constants $A > B$ such that $\frac{f\left( \frac{1}{1+2x}\right) }{f(x)}$ is independent of $x$ , where $f(x) = \frac{1 + Ax}{1 + Bx}$ for all real $x \ne - \frac{1}{B}$ . Put $a_0 = 1$ , $a_{n+1} = \frac{1}{1 + 2a_n}$ . Find an expression for an by considering $f(a_0), f(a_1), ...$ .