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

Given f(x+1)=2^(f(x)) .f(1) and f(1)= 16 then f(2016)=?

$$\:\:{Given}\:{f}\left({x}+\mathrm{1}\right)=\mathrm{2}^{{f}\left({x}\right)} .{f}\left(\mathrm{1}\right) \\ $$$$\:\:{and}\:{f}\left(\mathrm{1}\right)=\:\mathrm{16}\: \\ $$$$\:\:{then}\:{f}\left(\mathrm{2016}\right)=? \\ $$

Answered by Berbere last updated on 04/Apr/24

f(2)=2^(20) =2^2^4 f(3)=2^(2^(20) +4) f(4)=2^(2^(2^(20) +4) +4) g(n)=2^n h(n)=n+4; f(3)=gohog(20) f(4)=gohogohog(20);∀n≥3 f(n)=goh........ohog(20),(n−1)g appair proof for n=3 f(3)=gohog(20) True f(n)=goho.....og(20) f(n+1)=2^(goho....og(20)+4) goh....og(20)+4=h(goh....og(20)) f(n+1)=2^(hogoh.....g(20)) =goh(f(n)) ⇒f(n+1)=gohogoh.....og(20);g appair n=Times f(2016)=goh....og(20),g appair 2015 times

$${f}\left(\mathrm{2}\right)=\mathrm{2}^{\mathrm{20}} =\mathrm{2}^{\mathrm{2}^{\mathrm{4}} } \\ $$$${f}\left(\mathrm{3}\right)=\mathrm{2}^{\mathrm{2}^{\mathrm{20}} +\mathrm{4}} \\ $$$${f}\left(\mathrm{4}\right)=\mathrm{2}^{\mathrm{2}^{\mathrm{2}^{\mathrm{20}} +\mathrm{4}} +\mathrm{4}} \\ $$$${g}\left({n}\right)=\mathrm{2}^{{n}} \\ $$$${h}\left({n}\right)={n}+\mathrm{4}; \\ $$$${f}\left(\mathrm{3}\right)={gohog}\left(\mathrm{20}\right) \\ $$$${f}\left(\mathrm{4}\right)={gohogohog}\left(\mathrm{20}\right);\forall{n}\geqslant\mathrm{3} \\ $$$${f}\left({n}\right)={goh}........{ohog}\left(\mathrm{20}\right),\left({n}−\mathrm{1}\right){g}\:{appair}\: \\ $$$${proof}\: \\ $$$${for}\:{n}=\mathrm{3} \\ $$$${f}\left(\mathrm{3}\right)={gohog}\left(\mathrm{20}\right)\:{True} \\ $$$${f}\left({n}\right)={goho}.....{og}\left(\mathrm{20}\right) \\ $$$${f}\left({n}+\mathrm{1}\right)=\mathrm{2}^{{goho}....{og}\left(\mathrm{20}\right)+\mathrm{4}} \\ $$$${goh}....{og}\left(\mathrm{20}\right)+\mathrm{4}={h}\left({goh}....{og}\left(\mathrm{20}\right)\right) \\ $$$${f}\left({n}+\mathrm{1}\right)=\mathrm{2}^{{hogoh}.....{g}\left(\mathrm{20}\right)} ={goh}\left({f}\left({n}\right)\right) \\ $$$$\Rightarrow{f}\left({n}+\mathrm{1}\right)={gohogoh}.....{og}\left(\mathrm{20}\right);{g}\:{appair}\:{n}={Times} \\ $$$${f}\left(\mathrm{2016}\right)={goh}....{og}\left(\mathrm{20}\right),{g}\:{appair}\:\mathrm{2015}\:{times} \\ $$$$ \\ $$$$ \\ $$$$ \\ $$

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