> For the complete documentation index, see [llms.txt](https://markibrahim.gitbook.io/why-calculus-works/llms.txt). Markdown versions of documentation pages are available by appending `.md` to page URLs; this page is available as [Markdown](https://markibrahim.gitbook.io/why-calculus-works/goodies.md).

# 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]