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]