Question

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

Answer 1

$let\frac{1}{3\sqrt{2}}\sqrt{4-\frac{1}{3\sqrt{2}}.........}=x$

$So,$

$\frac{1}{3\sqrt{2}}\sqrt{4-x}=x$

$\frac{1}{18}(4-x)=x^2$

$18x^2+x-4=0$

$18x^2+9x-8x-4=0$

$9x(2x+1)-4(2x+1)=0$

$(2x+1)(9x-4)=0$

$x=\frac{-1}{2}andx=\frac{4}{9}$

$Therefore,x=\frac{4}{9}$

$So,$

$6+{\log }_{\frac{3}{2}}\left(\frac{4}{9}\right)=6-2=4$