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Template:Sequence of the Day for October 18

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Intended for: October 18, 2012

Timetable

  • First draft entered by Alonso del Arte on April 8, 2012
  • Draft reviewed by Daniel Forgues on October 18, 2012
  • Draft to be approved by September 18, 2012
Yesterday's SOTD * Tomorrow's SOTD

The line below marks the end of the <noinclude> ... </noinclude> section.



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.