| Since the golden ratio satisfies this equation, we can immediately deduce its continued fraction. Let φ be the golden ratio. We want the positive solution. So φ-1 = 1/φ > 0, so φ > 1. But this means φ-1 = 1/φ < 1, so 1 < φ < 2, and the first term in the continued fraction of φ is 1.
So now we want to write φ = 1 + 1/x. But this means x = 1/(φ-1) = φ, so φ = 1 + 1/φ. Expanding, we write
1
φ = 1 + ---------------------
1
1 + -----------------
1
1 + -------------
1
1 + ---------
1 + ...
| 
