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f(1/(1+2x))/f(x) indendent of x if f(x) = \frac{1 + Ax}{1 + Bx}

Source: 1965 Swedish Mathematical Competition p4

March 21, 2021

algebrafunctionrecurrence relationSequence

Problem Statement

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), ...