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Algebra Level 4

A sequence of functions {fn(x)}\{f_n(x) \} is defined recursively as follows:

f1(x)=x2+48,andfn+1(x)=x2+6fn(x)for n1.\large \begin{aligned} f_1(x) &= \sqrt {x^2 + 48}, \quad \text{and} \\ f_{n + 1}(x) &= \sqrt {x^2 + 6f_n(x)} \quad \text{for } n \geq 1. \end{aligned}.

For each positive integer nn, if there is only one real solution of the equation fn(x)=2xf_n(x) = 2x, find that solution.

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