Question

Find the value of $1+\frac{1}{1+\frac{1}{1+\frac{1}{1+\frac{1}{....}}}}$

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

Answer 1

Answer: (D)

$\frac{1+\sqrt{5}}{2}$

Take $x=1+\frac{1}{1+\frac{1}{1+\frac{1}{1+\frac{1}{....}}}}$
$\therefore x=1+\frac{1}{x}$
or $x=\frac{x+1}{x}$ or $x^2-x-1=0$
or $x=\frac{1\pm \sqrt{1+4}}{2}=\frac{1\pm \sqrt{5}}{2}$
Since x cannot be negative, $x=\frac{1+\sqrt{5}}{2}$.