# Goodies

## Golden Ratio

$$\sqrt{1 + \sqrt{1 + \sqrt{1 + \dots}}} = ?$$

How do we go about solving this problem? Set the value equal to x:

$$x = \sqrt{1 + \sqrt{1 + \sqrt{1 + \dots}}}$$

implying

$$x^2 =1 + \sqrt{1 + \sqrt{1 + \sqrt{1 + \dots}}} = 1 + x$$

Now have a quadratic equation we can solve: $$x^2 - x - 1 =0$$, implying one solution is

$$x = (1 + \sqrt{5}) /2$$, which amazingly is the **golden ratio**!

\[in progress]
