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Question Number 116085 by Ar Brandon last updated on 30/Sep/20

$$6+{\log }_{\frac{3}{2}}\left\{\frac{1}{3\sqrt{2}}\sqrt{4-\frac{1}{3\sqrt{2}}\sqrt{4-\frac{1}{3\sqrt{2}}\sqrt{4-\frac{1}{3\sqrt{2}}\cdot \cdot \cdot }}}\right\}=?$$

Commented by Dwaipayan Shikari last updated on 30/Sep/20

$$\sqrt{4-\frac{1}{3\sqrt{2}}\sqrt{4-\frac{1}{3\sqrt{2}}....}}=\mathrm{a}$$$$4-\frac{1}{3\sqrt{2}}\mathrm{a}={\mathrm{a}}^2$$$$3\sqrt{2}{\mathrm{a}}^2+\mathrm{a}-12\sqrt{2}=0$$$$\mathrm{a}=\frac{-1\pm \sqrt{1+288}}{6\sqrt{2}}=\frac{16}{6\sqrt{2}}=\frac{8}{3\sqrt{2}}$$$$6+{\log }_{\frac{3}{2}}\frac{1}{3\sqrt{2}}.\frac{8}{3\sqrt{2}}=6+{\log }_{\frac{3}{2}}\frac{4}{9}=4$$

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