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Question Number 205988 by cortano12 last updated on 04/Apr/24

$$Givenf(x+1)=2^{f\left(x\right)}.f\left(1\right)$$$$andf\left(1\right)=16$$$$thenf\left(2016\right)=?$$

Answered by Berbere last updated on 04/Apr/24

$$f\left(2\right)=2^{20}=2^{2^4}$$$$f\left(3\right)=2^{2^{20}+4}$$$$f\left(4\right)=2^{2^{2^{20}+4}+4}$$$$g\left(n\right)=2^n$$$$h\left(n\right)=n+4;$$$$f\left(3\right)=gohog\left(20\right)$$$$f\left(4\right)=gohogohog\left(20\right);\mathrm{\forall }n⩾3$$$$f\left(n\right)=goh........ohog\left(20\right),(n-1)gappair$$$$proof$$$$forn=3$$$$f\left(3\right)=gohog\left(20\right)True$$$$f\left(n\right)=goho.....og\left(20\right)$$$$f(n+1)=2^{goho....og\left(20\right)+4}$$$$goh....og\left(20\right)+4=h(goh....og\left(20\right))$$$$f(n+1)=2^{hogoh.....g\left(20\right)}=goh\left(f\left(n\right)\right)$$$$\Rightarrow f(n+1)=gohogoh.....og\left(20\right);gappairn=Times$$$$f\left(2016\right)=goh....og\left(20\right),gappair2015times$$$$$$$$$$$$$$

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