Recurring Styles #5 - Really a unique Solution?

Algebra Level 5

\[\large{f_{n+1}(x) = \sqrt{x^2 + 6f_n(x)}, \quad f_0(x) = 8}\]

Define a sequence of functions \(f_0 , f_1, f_2 , \ldots\) such that it satisfies the above conditions, for whole numbers \(n\), and real \(x\). For every positive integer \(n\), solve the equation \({f_n(x) = 2x}\), for \({x}\).

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