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Sequence of the Day for October 18

A092290: Decimal expansion of the [positive] solution to
n
x
= x  −  n
for
n = 7
.
7.887482193696061...
Since
n
x
= x  −  n
gives the quadratic equation
x 2  −  nx  −  n = 0
with roots
x  = 
n ±
2  n 2 + 4 n
2
 ,
for
n = 7
, the positive root is
x  = 
7 +
2  77
2
 .
Also, since we have
x = n +
n
x
, the continued fraction for the positive root
x
is
     
x  =  n + 
n
n + 
n
n + 
n
n + 
n
n + 
n
n + 
n
 .

Note that for
n = 1
, the positive root is
x  = 
1 +
2  5
2
 ,

which is the golden ratio.



The following is in the <noinclude> ... </noinclude> section.

For more details about today's Sequence of the Day, see {{Sequence of the Day for October 18}}.

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