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Question Number 153897 by mathdanisur last updated on 11/Sep/21

$$\mathrm{Denote}{\mathrm{x}}_{\mathbf{n}}\mathrm{is}\mathrm{the}\mathrm{unique}\mathrm{positive}\mathrm{root}$$$$\mathrm{of}\mathrm{the}\mathrm{following}\mathrm{equation}:$$$${\mathrm{x}}^{\mathbf{n}}+{\mathrm{x}}^{\mathbf{n}-1}+...\mathrm{x}=\mathrm{n}+2$$$$\mathrm{Prove}\mathrm{that}\mathrm{the}\mathrm{sequence}\left({\mathrm{x}}_{\mathbf{n}}\right)\mathrm{converges}$$$$\mathrm{to}\mathrm{a}\mathrm{positive}\mathrm{real}\mathrm{number}.\mathrm{Find}\mathrm{that}$$$$\mathrm{limit}.$$

Commented by MJS_new last updated on 12/Sep/21

$$\mathrm{this}\mathrm{is}\mathrm{not}\mathrm{a}\mathrm{proof}\mathrm{but}...$$$$f_n\left(x\right)=x^n+x^{n-1}+...+x$$$$\Rightarrow$$$$f_n\left(1\right)=n$$$$\Rightarrow$$$$f_n(1+k)>n\mathrm{\forall }k>0$$$$\Rightarrow$$$$\mathrm{\exists }k>0:f_n(1+k)=n+2$$$$\mathrm{obviously}$$$$n\to \mathrm{\infty }\Rightarrow k\to 0$$

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