Question Number 204664 by cortano12 last updated on 25/Feb/24
$$\:\:\mathrm{8}+\sqrt{\mathrm{8}^{\mathrm{2}} +\sqrt{\mathrm{8}^{\mathrm{4}} +\sqrt{\mathrm{8}^{\mathrm{8}} +\sqrt{\mathrm{8}^{\mathrm{16}} +\sqrt{...}}}}}\:=\:?\: \\ $$
Answered by Frix last updated on 25/Feb/24
$$=\mathrm{8}+\mathrm{8}\sqrt{\mathrm{1}+\sqrt{\mathrm{1}+\sqrt{\mathrm{1}+\sqrt{\mathrm{1}+...}}}} \\ $$$$=\mathrm{8}+\mathrm{8}\varphi=\mathrm{12}+\mathrm{4}\sqrt{\mathrm{5}} \\ $$
Commented by cortano12 last updated on 25/Feb/24
$$\varphi\:=\:\mathrm{golden}\:\mathrm{ratio}\: \\ $$
Answered by SEKRET last updated on 25/Feb/24
$$\:\:\mathrm{8}+\sqrt{\mathrm{8}\centerdot\mathrm{8}+\mathrm{8}\sqrt{\mathrm{8}^{\mathrm{2}} +\sqrt{\mathrm{8}^{\mathrm{4}} +\sqrt{\mathrm{8}^{\mathrm{8}} +....\:\:\:}}}}\:=\:\boldsymbol{\mathrm{X}} \\ $$$$\:\mathrm{8}+\sqrt{\mathrm{8}}\:\centerdot\sqrt{\mathrm{8}+\sqrt{\mathrm{8}^{\mathrm{2}} +\sqrt{\mathrm{8}^{\mathrm{4}} +...^{\:\:} }}\:}\:=\:\boldsymbol{\mathrm{X}} \\ $$$$\:\:\mathrm{8}+\sqrt{\mathrm{8}\:}\:\centerdot\sqrt{\boldsymbol{\mathrm{X}}}\:=\:\boldsymbol{\mathrm{X}} \\ $$$$\:\:\:\:\:\boldsymbol{\mathrm{X}}=\:\mathrm{12}+\mathrm{4}\sqrt{\mathrm{5}} \\ $$$$\:\:\:\: \\ $$