Question

The value of the continued fraction $1+\frac{1}{1+\frac{1}{1+\frac{1}{1+....\infty }}}$, is

• A. $\frac{\sqrt{5}-1}{2}$
• B. $\frac{\sqrt{5}+1}{2}$
• C. $\frac{\sqrt{5}-1}{4}$
• D. $\frac{\sqrt{5}+1}{4}$

Answer 1

The correct option is B $\frac{\sqrt{5}+1}{2}$

$y=1+\frac{1}{1+\frac{1}{1+\frac{1}{1+...\infty }}}$
Therefore
$y=1+\frac{1}{y}$
$y^2=y+1$
$y^2-y-1=0$
$y=\frac{1\pm \sqrt{1+4}}{2}$
$y=\frac{1+\sqrt{5}}{2}$ and $y=\frac{1-\sqrt{5}}{2}$
Since y is strictly posititve,
$y=\frac{\sqrt{5}+1}{2}$