Question Number 186739 by Davidtim last updated on 09/Feb/23
$$\sqrt[{\mathrm{3}}]{\mathrm{5}+\sqrt[{\mathrm{3}}]{\mathrm{5}+\sqrt[{\mathrm{3}}]{\mathrm{5}+…}}}=? \\ $$
Answered by pablo1234523 last updated on 09/Feb/23
$${x}=\sqrt[{\mathrm{3}}]{\mathrm{5}+\sqrt[{\mathrm{3}}]{\mathrm{5}+\sqrt[{\mathrm{3}}]{\mathrm{5}+…}}} \\ $$$$\Rightarrow{x}=\sqrt[{\mathrm{3}}]{\mathrm{5}+{x}} \\ $$$$\Rightarrow{x}^{\mathrm{3}} −{x}−\mathrm{5}=\mathrm{0} \\ $$$$\Rightarrow{x}\approx\mathrm{1}.\mathrm{9042}\:\left({real}\:{solution}\right) \\ $$
Commented by Frix last updated on 09/Feb/23
$${x}=\sqrt[{\mathrm{3}}]{\frac{\mathrm{5}}{\mathrm{2}}−\frac{\sqrt{\mathrm{2013}}}{\mathrm{18}}}+\sqrt[{\mathrm{3}}]{\frac{\mathrm{5}}{\mathrm{2}}+\frac{\sqrt{\mathrm{2013}}}{\mathrm{18}}} \\ $$