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]