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Question Number 111149 by Aina Samuel Temidayo last updated on 02/Sep/20

$$\mathrm{Let}{\mathrm{f}}_0\left(\mathrm{x}\right)=\frac{1}{1-\mathrm{x}}\mathrm{and}{\mathrm{f}}_{\mathrm{n}}\left(\mathrm{x}\right)$$$$={\mathrm{f}}_0\left({\mathrm{f}}_{\mathrm{n}-1}\left(\mathrm{x}\right)\right),\mathrm{n}=1,2,3,...\mathrm{Evaluate}$$$${\mathrm{f}}_{2018}\left(2018\right)$$

Commented by kaivan.ahmadi last updated on 02/Sep/20

$$f_1\left(x\right)=f_0\left(f_0\left(x\right)\right)=f_0\left(\frac{1}{1-x}\right)=\frac{1}{1-\frac{1}{1-x}}=\frac{1}{\frac{1-x-1}{1-x}}=\frac{1-x}{-x}=1-\frac{1}{x}$$$$f_2\left(x\right)=\frac{1}{1-1+\frac{1}{x}}=x$$$$f_3\left(x\right)=\frac{1}{1-x}=f_0\left(x\right)$$$$\Rightarrow f_n\left(x\right)=f_i;i=0or1or2andn\stackrel{3}{\equiv }i$$$$2018\stackrel{3}{\equiv }2\Rightarrow f_{2018}\left(x\right)=f_2\left(x\right)=x$$$$\Rightarrow f_{2018}\left(2018\right)=2018$$

Commented by Aina Samuel Temidayo last updated on 03/Sep/20

$$\mathrm{Please}\mathrm{I}{\mathrm{don}}^{\prime }\mathrm{t}\mathrm{understand}\mathrm{these}\mathrm{lines}.$$$$\mathrm{I}\mathrm{will}\mathrm{be}\mathrm{glad}\mathrm{if}\mathrm{you}\mathrm{can}\mathrm{help}\mathrm{shed}\mathrm{more}$$$$\mathrm{light}\mathrm{on}\mathrm{it}.$$$$\Rightarrow {\mathrm{f}}_{\mathrm{n}}\left(\mathrm{x}\right)={\mathrm{f}}_{\mathrm{i}},\mathrm{i}=0\mathrm{or}1\mathrm{or}2\mathrm{and}\mathrm{n}\stackrel{3}{\equiv }\mathrm{i}$$$$2018\stackrel{3}{\equiv }2\Rightarrow {\mathrm{f}}_{2018}\left(\mathrm{x}\right)={\mathrm{f}}_2\left(\mathrm{x}\right)=\mathrm{x}$$

Commented by kaivan.ahmadi last updated on 03/Sep/20

$$f_0\left(x\right)=\frac{1}{1-x}$$$$f_1\left(x\right)=1-\frac{1}{x}$$$$f_2\left(x\right)=x$$$$\Rightarrow f_3\left(x\right)=f_0\left(x\right)f_5\left(x\right)=f_1\left(x\right)f_6\left(x\right)=f_2\left(x\right)$$$$so{f}_n\left(x\right)isaoneofthe{f}_0,f_1,f_2$$$$f_n\left(x\right)=\{\begin{array}{l}{f}_0\left(x\right);n=3k\\ f_1\left(x\right);n=3k+1(k\in \mathbb{Z})\\ f_2\left(x\right);n=3k+2\end{array}$$$$nowsince2018=3\left(672\right)+2wehave$$$$f_{2018}\left(x\right)=f_2\left(x\right)=x$$$$$$$$$$$$$$

Commented by Aina Samuel Temidayo last updated on 03/Sep/20

$$\mathrm{I}\mathrm{understand}\mathrm{now}.\mathrm{Thanks}.$$

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