If $$X = \frac{1}{1+\frac{1}{1+X}}$$ and $$Y = \frac{2}{2+\frac{1}{1+Y}}$$, then which of the following can be the value of $$X+Y$$

$$X=\frac{1}{1+\frac{1}{1+X}}$$

or, $$X=\frac{1+X}{1+X+1}.$$

or, $$X\left(2+X\right)=1+X.$$

or, $$2X+X^2=1+X.$$

or, $$X^2+X-1=0.$$

or, $$X=\frac{-1\pm\ \sqrt{1^2+4}}{2.1}=\frac{-1\pm\sqrt{5}}{2}.$$

And, 

$$Y=\frac{2}{2+\frac{1}{1+Y}}.$$

or, $$Y=\frac{2+2Y}{2+2Y+1}.$$

or, $$Y\left(3+2Y\right)=2+2Y.$$

or, $$3Y+2Y^2=2+Y.$$

or, $$2Y^2+Y-2=0.$$

or, $$Y=\frac{-1\pm\sqrt{1+16}}{2.2}=\frac{-1\pm\sqrt{17}}{4}.$$

So, $$X+Y=\frac{-1+\sqrt{5}}{2}+\frac{-1+\sqrt{17}}{4}=\frac{-2+2\sqrt{5}-1+\sqrt{17}}{4}=\frac{-3+\sqrt{17}+2\sqrt{5}}{4}.$$(by taking positive roots.)

B is correct choice.