Question

The value of $y=2+\frac{1}{4+\frac{1}{4+\frac{1}{4+\frac{1}{4+......\infty }}}}$ is

• A. $6$
• B. $-\sqrt{5}$
• C. $\sqrt{5}$
• D. $\sqrt{6}$

Answer 1

The correct option is C $\sqrt{5}$

$y=2+\frac{1}{4+\frac{1}{4+\frac{1}{4+\frac{1}{4+......\infty }}}}\Rightarrow y+2=4+\frac{1}{4+\frac{1}{4+\frac{1}{4+\frac{1}{4+......\infty }}}}\Rightarrow y+2=4+\frac{1}{y+2}\Rightarrow (y+2{)}^2=4y+9\Rightarrow y^2-5=0\Rightarrow y=\pm \sqrt{5}$
As $y$ cannot be negative, so

Therefore, $y=\sqrt{5}$