Question Number 199801 by Calculusboy last updated on 09/Nov/23
Answered by Frix last updated on 09/Nov/23
$${x}={a}\sqrt{{b}−{a}\sqrt{{b}−{a}\sqrt{{b}−…}}} \\ $$$${x}={a}\sqrt{{b}−{x}} \\ $$$${x}^{\mathrm{2}} ={a}^{\mathrm{2}} \left({b}−{x}\right) \\ $$$${x}^{\mathrm{2}} +{a}^{\mathrm{2}} {x}−{a}^{\mathrm{2}} {b}=\mathrm{0} \\ $$$${x}=−\frac{{a}^{\mathrm{2}} }{\mathrm{2}}+\frac{{a}^{\mathrm{2}} \sqrt{{a}^{\mathrm{2}} +\mathrm{4}{b}}}{\mathrm{2}} \\ $$$${a}=\frac{\sqrt{\mathrm{2}}}{\mathrm{6}}\wedge{b}=\mathrm{4} \\ $$$${x}=\frac{\mathrm{4}}{\mathrm{9}} \\ $$$$\mathrm{6}+\frac{\mathrm{ln}\:\frac{\mathrm{4}}{\mathrm{9}}}{\mathrm{ln}\:\frac{\mathrm{3}}{\mathrm{2}}}=\mathrm{6}−\mathrm{2}=\mathrm{4} \\ $$
Commented by Calculusboy last updated on 10/Nov/23
$$\boldsymbol{{thanks}} \\ $$