Question Number 184503 by mr W last updated on 07/Jan/23
Answered by SEKRET last updated on 07/Jan/23
$$\:\:\boldsymbol{\mathrm{a}}\:=\sqrt{\mathrm{11}+\boldsymbol{\mathrm{a}}} \\ $$$$\:\:\:\boldsymbol{\mathrm{a}}^{\mathrm{2}} =\:\mathrm{11}\:+\:\boldsymbol{\mathrm{a}} \\ $$$$\:\:\:\boldsymbol{\mathrm{a}}^{\mathrm{3}} =\:\mathrm{11}\boldsymbol{\mathrm{a}}+\boldsymbol{\mathrm{a}}^{\mathrm{2}} =\mathrm{11}\boldsymbol{\mathrm{a}}+\mathrm{11}+\boldsymbol{\mathrm{a}}=\mathrm{12}\boldsymbol{\mathrm{a}}+\mathrm{11} \\ $$$$\:\:\:\boldsymbol{\mathrm{a}}^{\mathrm{4}} =\mathrm{12}\boldsymbol{\mathrm{a}}^{\mathrm{2}} +\mathrm{11}\boldsymbol{\mathrm{a}}=\mathrm{12}\centerdot\mathrm{11}+\mathrm{12}\boldsymbol{\mathrm{a}}+\mathrm{11}\boldsymbol{\mathrm{a}}=\mathrm{12}\centerdot\mathrm{11}+\mathrm{23}\boldsymbol{\mathrm{a}} \\ $$$$\:\boldsymbol{\mathrm{a}}^{\mathrm{4}} −\boldsymbol{\mathrm{a}}^{\mathrm{3}} −\mathrm{11}\boldsymbol{\mathrm{a}}^{\mathrm{2}} +\mathrm{11}= \\ $$$$\:\mathrm{12}\centerdot\mathrm{11}+\mathrm{23}\boldsymbol{\mathrm{a}}−\mathrm{12}\boldsymbol{\mathrm{a}}−\mathrm{11}−\mathrm{11}^{\mathrm{2}} −\mathrm{11}\boldsymbol{\mathrm{a}}+\mathrm{11}= \\ $$$$=\:\mathrm{12}\centerdot\mathrm{11}−\mathrm{11}^{\mathrm{2}} +\mathrm{23}\boldsymbol{\mathrm{a}}−\mathrm{23}\boldsymbol{\mathrm{a}}= \\ $$$$\:=\:\mathrm{11}\left(\mathrm{12}−\mathrm{11}\right)=\mathrm{1}\centerdot\mathrm{11}=\mathrm{11} \\ $$
Commented by SEKRET last updated on 07/Jan/23
$$\:\:\:\boldsymbol{\mathrm{a}}=\sqrt{\mathrm{11}+\sqrt{\mathrm{11}+\sqrt{\mathrm{11}+\sqrt{\mathrm{11}+\sqrt{\mathrm{11}+\sqrt{\mathrm{11}+\sqrt{\mathrm{11}+\mathrm{11}+\sqrt{\mathrm{11}+....}}}}}}}} \\ $$
Commented by mr W last updated on 07/Jan/23
$${a}^{\mathrm{2}} −{a}=\mathrm{11} \\ $$$${a}^{\mathrm{4}} −{a}^{\mathrm{3}} −\mathrm{11}{a}^{\mathrm{2}} +\mathrm{11} \\ $$$$={a}^{\mathrm{2}} \left({a}^{\mathrm{2}} −{a}\right)−\mathrm{11}{a}^{\mathrm{2}} +\mathrm{11} \\ $$$$=\mathrm{11}{a}^{\mathrm{2}} −\mathrm{11}{a}^{\mathrm{2}} +\mathrm{11} \\ $$$$=\mathrm{11} \\ $$