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Goodies
Golden Ratio
1
+
1
+
1
+
…
=
?
\sqrt{1 + \sqrt{1 + \sqrt{1 + \dots}}} = ?
1
+
1
+
1
+
…
=
?
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}}}
x
=
1
+
1
+
1
+
…
implying
x
2
=
1
+
1
+
1
+
1
+
…
=
1
+
x
x^2 =1 + \sqrt{1 + \sqrt{1 + \sqrt{1 + \dots}}} = 1 + x
x
2
=
1
+
1
+
1
+
1
+
…
=
1
+
x
Now have a quadratic equation we can solve:
x
2
−
x
−
1
=
0
x^2 - x - 1 =0
x
2
−
x
−
1
=
0
, implying one solution is
x
=
(
1
+
5
)
/
2
x = (1 + \sqrt{5}) /2
x
=
(
1
+
5
)
/2
, which amazingly is the
golden ratio
!
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