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Question Number 164299 by mr W last updated on 15/Jan/22

$$if\underset{ntimes}{f\left(f\left(f(...f\left(x\right)...)\right)\right)}=2x+1,$$$$findf\left(1\right)=?$$

Answered by ajfour last updated on 16/Jan/22

$$y_1=ax+b$$$$y_2=a(ax+b)+b$$$$y_3=a^2(ax+b)+ab+b$$$$y_n=a^{n}x+(a^{n-1}+a^{n-2}+...+a+1)b$$$$=2x+1$$$$\Rightarrow a^n=2$$$$\left(\frac{a^n-1}{a-1}\right)b=1$$$$\Rightarrow b=a-1=2^{1/n}-1$$$$y_1\left(1\right)=f\left(1\right)=a+b=2a-1$$$$\Rightarrow f\left(1\right)=2^{\frac{n+1}{n}}-1$$$$e.g.sayn=5$$$$y_5=a^{5}x+b\left(\frac{a^5-1}{a-1}\right)=2x+1$$$$\Rightarrow a^5=2;b=2^{1/5}-1$$$$y_1=\left(2^{1/5}\right)x+2^{1/5}-1$$$$y_1\left(1\right)=2^{6/5}-1\left(seeblueformula\right).$$

Commented by mr W last updated on 16/Jan/22

$$perfectsir!$$

Commented by Tawa11 last updated on 16/Jan/22

$$\mathrm{Great}\mathrm{sir}$$

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