Template:Sequence of the Day for October 18

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  −  n x  −  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.