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solutions of x = f_n(x), composition, f(x) = 1 + 2/x

Source: 1966 Swedish Mathematical Competition p4

March 21, 2021
algebracompositionSequence

Problem Statement

Let f(x)=1+2xf(x) = 1 + \frac{2}{x}. Put f1(x)=f(x)f_1(x) = f(x), f2(x)=f(f1(x))f_2(x) = f(f_1(x)), f3(x)=f(f2(x))f_3(x) = f(f_2(x)), ...... . Find the solutions to x=fn(x)x = f_n(x) for n>0n > 0.
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