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f f(x)=3x-2, then what is the value of x in the following equation? x=f(f(f(...f(x)...))) [2020 times]

saifmd wrote: ↑

Wed Mar 17, 2021 12:26 am

$f\circ f(x)=3x-2$, then what is the value of $x$ in the following equation? $x=\underbrace{f(f(f(\dots f(x)\dots)))}_{2020}$

Let's define $g$ such that $g(x)=f(f(x))$. Let's denote $\underbrace{g(g(\dots g(x)\dots))}_{n}$ by $g^n(x)$.
We can apply induction to get $g^n(x)=3^nx-3^n+1$.
Given,
$g^{1010}(x)=3^{1010}x-3^{1010}+1=x\Rightarrow \boxed{x=1}$