Find the value of
\(\cfrac{1}{1 + \cfrac{1}{2 + \cfrac{1}{1 + \cfrac{1}{2 + \dotsb}}}}\)
+506 First, set this equal to x:
\(x = \frac{1}{1+\frac{1}{2+\frac{1}{1+\frac{1}{2+...}}}}\)
We can replace the bottom part with x, shown below:
\(x = \frac{1}{1+\frac{1}{2+x}}\)
Now, solve for x:
\(x = \frac{1}{\frac{2+x+1}{2+x}}\\ x = \frac{1}{\frac{3+x}{2+x}}\\ x=\frac{2+x}{3+x}\\ x^2+3x=2+x\\ x^2+2x-2=0 \)
After using the quadratic formula or completing the square, we find that \(x = -1-\sqrt{3}\) or \(x = \sqrt{3}-1\). Since x must be positive, the negative solution is extraneous, and therefore the answer is \(\boxed{\sqrt{3}-1}\)
