Continued Fractions

Many mathematicians know that: $$\phi = 1 + \cfrac{1}{1 + \cfrac{1}{1 + \cfrac{1}{1 + \cfrac{1}{1 + \ddots}}}}$$ as a continued fraction. But do you know why?

Here are some problems on the theme 'continued':

1) Evaluate $\sqrt{3\sqrt{3\sqrt{3\cdots}}}$

2) Evaluate $\sqrt{3+\sqrt{3+\sqrt{3+\cdots}}}$

3) Evaluate $\frac{\frac{\frac{a+1}{2}+1}{2}+1}{2}+1\cdots$