AoPS Wiki talk:Problem of the Day/September 10, 2011
Revision as of 08:24, 10 September 2011 by Negativebplusorminus (talk | contribs)
Solution
Let
![\[S=\frac{F_1}{3}+\frac{F_2}{9}+\frac{F_3}{27}+\frac{F_4}{81}+\frac{F_5}{243}+\cdots\]](http://latex.artofproblemsolving.com/0/a/c/0aceefaa8437fe59a614d66111dae99ff68739f1.png)
Multiplying through by 1/3, we see
![\[\frac{1}{3}S=\frac{F_1}{9}+\frac{F_2}{27}+\frac{F_3}{81}+\frac{F_4}{243}+\cdots\]](http://latex.artofproblemsolving.com/8/c/d/8cdc55ae6087007c92b7e7bcd4d3bd2b34b9a32c.png)
and, noting that 
, we subtract:
![\[\frac{2}{3}S=\frac{1}{3}+\frac{F_1}{27}+\frac{F_2}{81}+\frac{F_3}{243}+\cdots\]](http://latex.artofproblemsolving.com/c/2/5/c25fe48a1aa3c4cd6ca866ee0355c9f311f080e4.png)
and, multiplying through by 9, we see S in the right side:
![\[6S=3+\frac{F_1}{3}+\frac{F_2}{9}+\frac{F_3}{27}+\frac{F_4}{81}+\frac{F_5}{243}+\cdots=3+S\]](http://latex.artofproblemsolving.com/d/1/c/d1caadd75f9b1a82652e005cb9cdf73761070422.png)
and thus 
and 
.
