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Question Number 97428 by bobhans last updated on 08/Jun/20

Commented by Farruxjano last updated on 08/Jun/20

I could not know x=1005 or not in my solution if so i think you can easily calculate removing x with 1005, sir

$$\boldsymbol{{I}}\:\boldsymbol{{could}}\:\boldsymbol{{not}}\:\boldsymbol{{know}}\:\boldsymbol{{x}}=\mathrm{1005}\:\boldsymbol{{or}}\:\boldsymbol{{not}}\:\boldsymbol{{in}} \\ $$$$\boldsymbol{{my}}\:\boldsymbol{{solution}}\:\boldsymbol{{if}}\:\boldsymbol{{so}}\:\boldsymbol{{i}}\:\boldsymbol{{think}}\:\boldsymbol{{you}} \\ $$$$\boldsymbol{{can}}\:\boldsymbol{{easily}}\:\boldsymbol{{calculate}}\:\boldsymbol{{removing}}\: \\ $$$$\boldsymbol{{x}}\:\boldsymbol{{with}}\:\mathrm{1005},\:\boldsymbol{{sir}} \\ $$$$ \\ $$

Commented by bobhans last updated on 08/Jun/20

sorry sir. i meant the question f(f(f(f(....))))(1005)

$$\mathrm{sorry}\:\mathrm{sir}.\:\mathrm{i}\:\mathrm{meant}\:\mathrm{the}\:\mathrm{question} \\ $$$$\mathrm{f}\left(\mathrm{f}\left(\mathrm{f}\left(\mathrm{f}\left(....\right)\right)\right)\right)\left(\mathrm{1005}\right) \\ $$

Commented by bemath last updated on 08/Jun/20

f(f(x)) = ((x/(√(1+x^2 )))/(√(1+(x^2 /(1+x^2 ))))) = (x/(√(1+2x^2 ))) f(f(f(x)))= ((x/(√(1+2x^2 )))/(√(1+(x^2 /(1+2x^2 ))))) = (x/(√(1+3x^2 ))) so so f(f(f(f(...))))(1005) = ((1005)/(√(1+1005×1005^2 ))) = ((1005)/(√(1+1005^3 )))

$$\boldsymbol{\mathrm{f}}\left(\boldsymbol{\mathrm{f}}\left(\boldsymbol{\mathrm{x}}\right)\right)\:=\:\frac{\frac{\boldsymbol{\mathrm{x}}}{\sqrt{\mathrm{1}+\boldsymbol{\mathrm{x}}^{\mathrm{2}} }}}{\sqrt{\mathrm{1}+\frac{\boldsymbol{\mathrm{x}}^{\mathrm{2}} }{\mathrm{1}+\boldsymbol{\mathrm{x}}^{\mathrm{2}} }}}\:=\:\frac{\boldsymbol{\mathrm{x}}}{\sqrt{\mathrm{1}+\mathrm{2}\boldsymbol{\mathrm{x}}^{\mathrm{2}} }} \\ $$$$\boldsymbol{\mathrm{f}}\left(\boldsymbol{\mathrm{f}}\left(\boldsymbol{\mathrm{f}}\left(\boldsymbol{\mathrm{x}}\right)\right)\right)=\:\frac{\frac{\boldsymbol{\mathrm{x}}}{\sqrt{\mathrm{1}+\mathrm{2}\boldsymbol{\mathrm{x}}^{\mathrm{2}} }}}{\sqrt{\mathrm{1}+\frac{\boldsymbol{\mathrm{x}}^{\mathrm{2}} }{\mathrm{1}+\mathrm{2}\boldsymbol{\mathrm{x}}^{\mathrm{2}} }}}\:=\:\frac{\boldsymbol{\mathrm{x}}}{\sqrt{\mathrm{1}+\mathrm{3}\boldsymbol{\mathrm{x}}^{\mathrm{2}} }} \\ $$$$\boldsymbol{\mathrm{so}}\:\boldsymbol{\mathrm{so}}\:\boldsymbol{\mathrm{f}}\left(\boldsymbol{\mathrm{f}}\left(\boldsymbol{\mathrm{f}}\left(\boldsymbol{\mathrm{f}}\left(...\right)\right)\right)\right)\left(\mathrm{1005}\right)\:= \\ $$$$\frac{\mathrm{1005}}{\sqrt{\mathrm{1}+\mathrm{1005}×\mathrm{1005}^{\mathrm{2}} }}\:=\:\frac{\mathrm{1005}}{\sqrt{\mathrm{1}+\mathrm{1005}^{\mathrm{3}} }} \\ $$

Commented by 1549442205 last updated on 08/Jun/20

your solution is nice greatly!it is followed by mathematical induction argument

$$\mathrm{your}\:\mathrm{solution}\:\mathrm{is}\:\mathrm{nice}\:\mathrm{greatly}!\mathrm{it}\:\mathrm{is}\:\mathrm{followed} \\ $$$$\mathrm{by}\:\mathrm{mathematical}\:\mathrm{induction}\:\mathrm{argument} \\ $$

Commented by bobhans last updated on 08/Jun/20

thank you

$$\mathrm{thank}\:\mathrm{you} \\ $$

Answered by Farruxjano last updated on 08/Jun/20

f(x)^1 =(√(x^2 /(1+x^2 ))) f(f(x))^2 =(√((x^2 /(x^2 +1))/(1+(x^2 /(x^2 +1)))))=(√(x^2 /(2x^2 +1))) f(f(f(x)))^3 =(√((x^2 /(x^2 +1))/(((2x^2 )/(x^2 +1))+1)))=(√(x^2 /(3x^2 +1))) ....................................................... f(f(...(f(x)))...)^(1005) =(√(x^2 /(1005x^2 +1)))

$$\boldsymbol{{f}}\left(\boldsymbol{{x}}\right)^{\mathrm{1}} =\sqrt{\frac{\boldsymbol{{x}}^{\mathrm{2}} }{\mathrm{1}+\boldsymbol{{x}}^{\mathrm{2}} }}\: \\ $$$$\boldsymbol{{f}}\left(\boldsymbol{{f}}\left(\boldsymbol{{x}}\right)\right)^{\mathrm{2}} =\sqrt{\frac{\frac{\boldsymbol{{x}}^{\mathrm{2}} }{\boldsymbol{{x}}^{\mathrm{2}} +\mathrm{1}}}{\mathrm{1}+\frac{\boldsymbol{{x}}^{\mathrm{2}} }{\boldsymbol{{x}}^{\mathrm{2}} +\mathrm{1}}}}=\sqrt{\frac{\boldsymbol{{x}}^{\mathrm{2}} }{\mathrm{2}\boldsymbol{{x}}^{\mathrm{2}} +\mathrm{1}}} \\ $$$$\boldsymbol{{f}}\left(\boldsymbol{{f}}\left(\boldsymbol{{f}}\left(\boldsymbol{{x}}\right)\right)\right)^{\mathrm{3}} =\sqrt{\frac{\frac{\boldsymbol{{x}}^{\mathrm{2}} }{\boldsymbol{{x}}^{\mathrm{2}} +\mathrm{1}}}{\frac{\mathrm{2}\boldsymbol{{x}}^{\mathrm{2}} }{\boldsymbol{{x}}^{\mathrm{2}} +\mathrm{1}}+\mathrm{1}}}=\sqrt{\frac{\boldsymbol{{x}}^{\mathrm{2}} }{\mathrm{3}\boldsymbol{{x}}^{\mathrm{2}} +\mathrm{1}}} \\ $$$$....................................................... \\ $$$$\boldsymbol{{f}}\left(\boldsymbol{{f}}\left(...\left(\boldsymbol{{f}}\left(\boldsymbol{{x}}\right)\right)\right)...\right)^{\mathrm{1005}} =\sqrt{\frac{\boldsymbol{{x}}^{\mathrm{2}} }{\mathrm{1005}\boldsymbol{{x}}^{\mathrm{2}} +\mathrm{1}}} \\ $$

Answered by mathmax by abdo last updated on 08/Jun/20

f(x) =(x/(√(1+x^2 ))) ⇒f^2 (x) =fof(x) =((f(x))/(√(1+f^2 (x)))) =((x/(√(1+x^2 )))/(√(1+(x^2 /(1+x^2 ))))) =(x/(√(1+2x^2 ))) let suppose f0f0...0f(x)(ntime) =(x/(√(1+nx^2 ))) ⇒ fofof...of(n+1)time =f((x/(√(1+nx^2 )))) =((x/(√(1+nx^2 )))/(√(1+(x^2 /(1+nx^2 ))))) =(x/(√(1+(n+1)x^2 ))) the relation is true at term(n+1) ⇒fofo...of(1005time) =(x/(√(1+(1006)x^2 )))

$$\mathrm{f}\left(\mathrm{x}\right)\:=\frac{\mathrm{x}}{\sqrt{\mathrm{1}+\mathrm{x}^{\mathrm{2}} }}\:\Rightarrow\mathrm{f}^{\mathrm{2}} \left(\mathrm{x}\right)\:=\mathrm{fof}\left(\mathrm{x}\right)\:=\frac{\mathrm{f}\left(\mathrm{x}\right)}{\sqrt{\mathrm{1}+\mathrm{f}^{\mathrm{2}} \left(\mathrm{x}\right)}}\:=\frac{\frac{\mathrm{x}}{\sqrt{\mathrm{1}+\mathrm{x}^{\mathrm{2}} }}}{\sqrt{\mathrm{1}+\frac{\mathrm{x}^{\mathrm{2}} }{\mathrm{1}+\mathrm{x}^{\mathrm{2}} }}}\:=\frac{\mathrm{x}}{\sqrt{\mathrm{1}+\mathrm{2x}^{\mathrm{2}} }} \\ $$$$\mathrm{let}\:\mathrm{suppose}\:\mathrm{f0f0}...\mathrm{0f}\left(\mathrm{x}\right)\left(\mathrm{ntime}\right)\:=\frac{\mathrm{x}}{\sqrt{\mathrm{1}+\mathrm{nx}^{\mathrm{2}} }}\:\Rightarrow \\ $$$$\mathrm{fofof}...\mathrm{of}\left(\mathrm{n}+\mathrm{1}\right)\mathrm{time}\:=\mathrm{f}\left(\frac{\mathrm{x}}{\sqrt{\mathrm{1}+\mathrm{nx}^{\mathrm{2}} }}\right)\:=\frac{\frac{\mathrm{x}}{\sqrt{\mathrm{1}+\mathrm{nx}^{\mathrm{2}} }}}{\sqrt{\mathrm{1}+\frac{\mathrm{x}^{\mathrm{2}} }{\mathrm{1}+\mathrm{nx}^{\mathrm{2}} }}}\:=\frac{\mathrm{x}}{\sqrt{\mathrm{1}+\left(\mathrm{n}+\mathrm{1}\right)\mathrm{x}^{\mathrm{2}} }} \\ $$$$\mathrm{the}\:\mathrm{relation}\:\mathrm{is}\:\mathrm{true}\:\mathrm{at}\:\mathrm{term}\left(\mathrm{n}+\mathrm{1}\right)\:\Rightarrow\mathrm{fofo}...\mathrm{of}\left(\mathrm{1005time}\right) \\ $$$$=\frac{\mathrm{x}}{\sqrt{\mathrm{1}+\left(\mathrm{1006}\right)\mathrm{x}^{\mathrm{2}} }} \\ $$

Commented by mathmax by abdo last updated on 08/Jun/20

sorry fofofo...of(1005time) =(x/(√(1+1005x^2 )))

$$\mathrm{sorry}\:\mathrm{fofofo}...\mathrm{of}\left(\mathrm{1005time}\right)\:=\frac{\mathrm{x}}{\sqrt{\mathrm{1}+\mathrm{1005x}^{\mathrm{2}} }} \\ $$

Commented by abdomathmax last updated on 08/Jun/20

⇒fof0...0f(1005)_(1005time) =((1005)/(√(1+(1005)^3 )))

$$\Rightarrow\mathrm{fof0}...\mathrm{0f}\left(\mathrm{1005}\right)_{\mathrm{1005time}} =\frac{\mathrm{1005}}{\sqrt{\mathrm{1}+\left(\mathrm{1005}\right)^{\mathrm{3}} }} \\ $$

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