Question Number 144474 by SOMEDAVONG last updated on 26/Jun/21
$$\mathrm{A}=\sqrt{\mathrm{1}+\sqrt{\mathrm{7}+\sqrt{\mathrm{1}+\sqrt{\mathrm{7}+\sqrt{\mathrm{1}+\sqrt{\mathrm{7}+...........}}}}}} \\ $$
Answered by mr W last updated on 25/Jun/21
$${x}=\sqrt{\mathrm{1}+\sqrt{\mathrm{7}+{x}}} \\ $$$$\left({x}^{\mathrm{2}} −\mathrm{1}\right)^{\mathrm{2}} =\mathrm{7}+{x} \\ $$$${x}^{\mathrm{4}} −\mathrm{2}{x}^{\mathrm{2}} −{x}−\mathrm{6}=\mathrm{0} \\ $$$$\left({x}−\mathrm{2}\right)\left({x}^{\mathrm{3}} +\mathrm{2}{x}^{\mathrm{2}} +\mathrm{2}{x}+\mathrm{3}\right)=\mathrm{0} \\ $$$$\Rightarrow{x}=\mathrm{2}>\mathrm{0} \\ $$