Question

The value of$y=6+{\log }_{3/2}\left[\frac{1}{3\sqrt{2}}\sqrt{4-\frac{1}{3\sqrt{2}}\sqrt{4-\frac{1}{3\sqrt{2}}\sqrt{4-\frac{1}{3\sqrt{2}}\cdots ..}}}\right]$ is (a) 2 (b) 3 (c) 4 (d) 8

• A. 2
• B. 3
• C. 4
• D. 8

Answer 1

Answer: (C)

Let $x=$
$\frac{1}{3\sqrt{2}}\sqrt{4-\frac{1}{3\sqrt{2}}\sqrt{4-\frac{1}{3\sqrt{2}}\sqrt{4-\frac{1}{3\sqrt{2}}\cdots ..}}}$
$\text{ then }x=\frac{1}{3\sqrt{2}}\sqrt{4-x}$
$\Rightarrow 18x^2+x-4=0$
$\Rightarrow (9x-4)(2x+1)=0$
$\Rightarrow x=4/9,-1/2$
$\text{ As }x>0\text{, we get }x=4/9$
$\text{ Thus, }y=6+{\log }_{3/2}(4/9)$
$=6+{\log }_{3/2}(3/2{)}^{-2}=4$