Nested Relations

$\begin{aligned} x & = \frac{1}{y+\frac{1}{y+\frac{1}{y+\frac{1}{...}}}} = \frac{1}{y + x} \quad \Rightarrow y + x = \frac{1}{x} \quad \Rightarrow \color{#3D99F6}{y = \frac{1}{x} - x} \\ y & = \sqrt{x+\sqrt{x+\sqrt{x+ ...}}} = \sqrt{x + y} \quad \Rightarrow \color{#D61F06}{y^2} = x + \color{#3D99F6}{y} = x + \color{#3D99F6}{\frac{1}{x} - x} = \color{#D61F06}{\frac{1}{x}} \end{aligned}$

$\begin{aligned} \left( \dfrac{1}{\color{#D61F06}{y}^{\color{#D61F06}{2}(2014)}} + x^{2014} \right) \left( \dfrac{1}{x^{2014}} + \color{#D61F06}{y}^{\color{#D61F06}{2}(2014)} \right) & = \left( \color{#D61F06}{x}^{2014} + x^{2014} \right) \left( \dfrac{1}{x^{2014}} + \dfrac{1}{\color{#D61F06}{x}^{2014}} \right) \\ & = 2x^{2014} \left( \dfrac{2}{x^{2014}} \right) = \boxed{4} \end{aligned}$

We can simplify the continued fraction as

$x = \large \frac {1}{y+x}$

while we can reduce the nested radical to the form

$y^{2} = \large x+y$

combining the two equations we get that

$x = \frac {1}{y^{2}}$

Now, simplifying the given expression, we get

$\left[\left(\frac{1}{xy^2}\right)^{2014}+\left(xy^2\right)^{2014}+2\right]$

Thus, substituting $xy^{2} = 1$ gives us a result of $\boxed {4}$ .

y^2= x+ y= (1/x)= K so, from given equation, ((1/y)^(2014)+ x^(2014))* ((1/x)^(2014)+ y^(2014))= ((1/K)^2014+ (1/K)^2014)* (K^2014+ K^2014) =( 2 (1/K)^2014) (2*K^2014) =4