Goodies

Golden Ratio

1+1+1+=?\sqrt{1 + \sqrt{1 + \sqrt{1 + \dots}}} = ?
How do we go about solving this problem? Set the value equal to x:
x=1+1+1+x = \sqrt{1 + \sqrt{1 + \sqrt{1 + \dots}}}
implying
x2=1+1+1+1+=1+xx^2 =1 + \sqrt{1 + \sqrt{1 + \sqrt{1 + \dots}}} = 1 + x
Now have a quadratic equation we can solve:
x2x1=0x^2 - x - 1 =0
, implying one solution is
x=(1+5)/2x = (1 + \sqrt{5}) /2
, which amazingly is the golden ratio!
[in progress]
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