Question

The value of $6+{\log }_{3/2}\left(\frac{1}{3\sqrt{2}}\sqrt{4-\frac{1}{3\sqrt{2}}\sqrt{4-\frac{1}{3\sqrt{2}}\sqrt{4-\frac{1}{3\sqrt{2}}\cdots }}}\right)$ is

Answer 1

Answer: 4

Let $\sqrt{4-\frac{1}{3\sqrt{2}}\sqrt{4-\frac{1}{3\sqrt{2}}\sqrt{4-\frac{1}{3\sqrt{2}}\cdots }}}=y$
So, $4-\frac{1}{3\sqrt{2}}y=y^2$
$(y>0)$
$\Rightarrow y^2+\frac{1}{3\sqrt{2}}y-4=0\Rightarrow y=\frac{8}{3\sqrt{2}}$
so, the required value is $6+{\log }_{3/2}\left(\frac{1}{3\sqrt{2}}\times \frac{8}{3\sqrt{2}}\right)$
$=6+{\log }_{3/2}\frac{4}{9}=6-2=4$