The golden ????o

Let $x$ be the infinite continued fraction $[1;1,1,1\cdots]$ Thus we get, $x=1+\dfrac{1}{x} \implies x^2-x-1=0$ Solving this using quadratic formula, you will get $x=\dfrac{1+\sqrt{(-1)^2-(4\times1\times-1)}}{2} = \dfrac{1+\sqrt{5}}{2}$ Or the golden ratio $\approx 1.618$

Edit: Since there are two values for the quadratic equation, we neglect the negative one since it is clear that the original equation is positive.