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

Commented by Farruxjano last updated on 08/Jun/20

$$\mathit{I}\mathit{c}\mathit{o}\mathit{u}\mathit{l}\mathit{d}\mathit{n}\mathit{o}\mathit{t}\mathit{k}\mathit{n}\mathit{o}\mathit{w}\mathit{x}=1005\mathit{o}\mathit{r}\mathit{n}\mathit{o}\mathit{t}\mathit{i}\mathit{n}$$$$\mathit{m}\mathit{y}\mathit{s}\mathit{o}\mathit{l}\mathit{u}\mathit{t}\mathit{i}\mathit{o}\mathit{n}\mathit{i}\mathit{f}\mathit{s}\mathit{o}\mathit{i}\mathit{t}\mathit{h}\mathit{i}\mathit{n}\mathit{k}\mathit{y}\mathit{o}\mathit{u}$$$$\mathit{c}\mathit{a}\mathit{n}\mathit{e}\mathit{a}\mathit{s}\mathit{i}\mathit{l}\mathit{y}\mathit{c}\mathit{a}\mathit{l}\mathit{c}\mathit{u}\mathit{l}\mathit{a}\mathit{t}\mathit{e}\mathit{r}\mathit{e}\mathit{m}\mathit{o}\mathit{v}\mathit{i}\mathit{n}\mathit{g}$$$$\mathit{x}\mathit{w}\mathit{i}\mathit{t}\mathit{h}1005,\mathit{s}\mathit{i}\mathit{r}$$$$$$

Commented by bobhans last updated on 08/Jun/20

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

Commented by bemath last updated on 08/Jun/20

$$\mathbf{f}\left(\mathbf{f}\left(\mathbf{x}\right)\right)=\frac{\frac{\mathbf{x}}{\sqrt{1+{\mathbf{x}}^2}}}{\sqrt{1+\frac{{\mathbf{x}}^2}{1+{\mathbf{x}}^2}}}=\frac{\mathbf{x}}{\sqrt{1+2{\mathbf{x}}^2}}$$$$\mathbf{f}\left(\mathbf{f}\left(\mathbf{f}\left(\mathbf{x}\right)\right)\right)=\frac{\frac{\mathbf{x}}{\sqrt{1+2{\mathbf{x}}^2}}}{\sqrt{1+\frac{{\mathbf{x}}^2}{1+2{\mathbf{x}}^2}}}=\frac{\mathbf{x}}{\sqrt{1+3{\mathbf{x}}^2}}$$$$\mathbf{so}\mathbf{so}\mathbf{f}\left(\mathbf{f}\left(\mathbf{f}\left(\mathbf{f}(...)\right)\right)\right)\left(1005\right)=$$$$\frac{1005}{\sqrt{1+1005\times {1005}^2}}=\frac{1005}{\sqrt{1+{1005}^3}}$$

Commented by 1549442205 last updated on 08/Jun/20

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

Commented by bobhans last updated on 08/Jun/20

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

Answered by Farruxjano last updated on 08/Jun/20

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

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

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

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

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

Commented by abdomathmax last updated on 08/Jun/20

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

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