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Question Number 107484 by bemath last updated on 11/Aug/20

$$\nparallel \mathcal{B}e\mathcal{M}ath\nparallel$$$$\sqrt{x+\sqrt{x+\sqrt{x+\sqrt{x+...}}}}=\sqrt{4\sqrt{4\sqrt{4\sqrt{4...}}}}$$$$x=?$$

Answered by john santu last updated on 11/Aug/20

$$\divideontimes \mathcal{JS}\divideontimes$$$$\Rightarrow x+\sqrt{x+\sqrt{x+\sqrt{x+...}}}=4\sqrt{4\sqrt{4\sqrt{4\sqrt{4...}}}}$$$$\Rightarrow x+\sqrt{4\sqrt{4\sqrt{4\sqrt{4\sqrt{4...}}}}}=4\sqrt{4\sqrt{4\sqrt{4\sqrt{4\sqrt{...}}}}}$$$$\left(1\right)lett=\sqrt{4\sqrt{4\sqrt{4\sqrt{4\sqrt{4\sqrt{4...}}}}}}$$$$t^2=4t\Rightarrow t=4$$$$\left(2\right)x+4=4.4\Rightarrow x=12$$

Answered by Dwaipayan Shikari last updated on 11/Aug/20

$$\sqrt{x+\sqrt{x+\sqrt{x+\sqrt{x....}}}}=p$$$$x+p=p^2$$$$p^2-p-x=0$$$$p=\frac{1+\sqrt{1+4x}}{2}$$$$\sqrt{4\sqrt{4\sqrt{4\sqrt{4\sqrt{4\sqrt{4}}}}}}=a$$$$4a=a^2$$$$a=4$$$$1+\sqrt{1+4x}=8$$$$1+4x=49$$$$x=12$$

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