2001 SMT/Algebra Problems/Problem 8

Problem

Determine the value of $1+\frac{1}{2+\frac{1}{1+\frac{1}{2+\frac{1}{1+\cdots}}}}$

Solution

Let $x=1+\frac{1}{2+\frac{1}{1+\frac{1}{2+\frac{1}{1+\cdots}}}}$ . Then by substitution: $x=1+\frac{1}{2+\frac{1}{x}}$ Simplifying, we find that $\frac{1}{x-1}=2+\frac{1}{x}$ , so $2x^2-2x-1=0$ . Thus $x=\frac{1\pm\sqrt{3}}{2}$ . Clearly $x$ is positive, so $x=\boxed{\frac{1+\sqrt{3}}{2}}$ .

~ eevee9406

Page

Toolbox

Search

2001 SMT/Algebra Problems/Problem 8

Problem

Solution