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Question Number 176850 by Ar Brandon last updated on 27/Sep/22

f(x)=2(x−1)  (fofo...of)_(50 times) (x)=?

$${f}\left({x}\right)=\mathrm{2}\left({x}−\mathrm{1}\right) \\ $$$$\underset{\mathrm{50}\:\mathrm{times}} {\underbrace{\left({fofo}...{of}\right)}}\left({x}\right)=? \\ $$

Answered by mr W last updated on 27/Sep/22

f(x)=2x−2  f(f(x))=2(2x−2)−2=2^2 x−(2^2 +2)  generally  f(f(f(...f(x))))=2^n x−(2^n +2^(n−1) +...+2^2 +2)  f(f(f(...f(x))))_(n times) =2^n x−2^(n+1) +2

$${f}\left({x}\right)=\mathrm{2}{x}−\mathrm{2} \\ $$$${f}\left({f}\left({x}\right)\right)=\mathrm{2}\left(\mathrm{2}{x}−\mathrm{2}\right)−\mathrm{2}=\mathrm{2}^{\mathrm{2}} {x}−\left(\mathrm{2}^{\mathrm{2}} +\mathrm{2}\right) \\ $$$${generally} \\ $$$${f}\left({f}\left({f}\left(...{f}\left({x}\right)\right)\right)\right)=\mathrm{2}^{{n}} {x}−\left(\mathrm{2}^{{n}} +\mathrm{2}^{{n}−\mathrm{1}} +...+\mathrm{2}^{\mathrm{2}} +\mathrm{2}\right) \\ $$$$\underset{{n}\:{times}} {{f}\left({f}\left({f}\left(...{f}\left({x}\right)\right)\right)\right)}=\mathrm{2}^{{n}} {x}−\mathrm{2}^{{n}+\mathrm{1}} +\mathrm{2} \\ $$

Commented by Ar Brandon last updated on 27/Sep/22

Thank you, Sir!

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