Infinite Fractions - 1

Let $\color{#3D99F6}{x}$ equal to the given expression.

$\Rightarrow \frac { 2 }{ 2+12\left( \frac { 2 }{ 2+12\left( \frac { 2 }{ 2+\dots } \right) } \right) } =\color{#3D99F6}{x}$

$\color{#3D99F6}{x}=\dfrac{2}{2+12(\color{#3D99F6}{x})}$

$\color{#3D99F6}{x}=\dfrac{2}{2(1+6\color{#3D99F6}{x})}$

$\color{#3D99F6}{x}=\dfrac{1}{1+6\color{#3D99F6}{x}}$

$6\color{#3D99F6}{x}^2+\color{#3D99F6}{x}-1=0$

By the Quadratic formula ,

$\color{#3D99F6}{x}=0.333, -0.5$

Because given expression consist only positive terms.Hence, neglecting $-0.5$ .

$\Rightarrow \frac { 2 }{ 2+12\left( \frac { 2 }{ 2+12\left( \frac { 2 }{ 2+\dots } \right) } \right) } = \color{#3D99F6}{0.333}$