+0

# help pls

0
56
1

Find the value of
$$\cfrac{1}{1 + \cfrac{1}{2 + \cfrac{1}{1 + \cfrac{1}{2 + \dotsb}}}}$$

Jan 18, 2021

### 1+0 Answers

#1
+168
0

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 $$\boxed{\sqrt{3}-1}$$

Jan 18, 2021