Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

Difference between revisions of "2007 Alabama ARML TST Problems/Problem 9"

(New page: ==Problem== Let <math>F_1=F_2=1</math>, and <math>F_{n+2}=F_{n+1}+F_n</math> for <math>n\geq 1</math>. Find the value of <math>k</math> such that <math>x=F_k</math> is the <math>x</math>-c...)
 
m (→‎Solution)
 
(One intermediate revision by the same user not shown)
Line 15: Line 15:
 
We can see the pattern:
 
We can see the pattern:
  
−
<math>F_{n}x+F_{n+1}y=-1^{n-1}F_{2011-n}</math>
+
<math>F_{n}x+F_{n+1}y=(-1)^{n-1}F_{2011-n}</math>
  
 
Thus
 
Thus
Line 26: Line 26:
  
 
==See also==
 
==See also==
−
(put box here)
+
{{ARML box|year=2007|state=Alabama|num-b=8|num-a=10}}
  
 
[[Category:Introductory Algebra Problems]]
 
[[Category:Introductory Algebra Problems]]

Latest revision as of 08:29, 27 August 2008

Problem

Let $F_1=F_2=1$, and $F_{n+2}=F_{n+1}+F_n$ for $n\geq 1$. Find the value of $k$ such that $x=F_k$ is the $x$-coordinate of the intersection between the linear equations

\[F_{2007} x + F_{2008} y = F_4,\]

\[F_{2008}x+F_{2009}y=-F_3.\]

Solution

We subtract equations:

$F_{2006}x+F_{2007}y=-F_5$

$F_{2005}x+F_{2006}y=F_6$

We can see the pattern:

$F_{n}x+F_{n+1}y=(-1)^{n-1}F_{2011-n}$

Thus

$x+y=F_{2010}$

$x+2y=-F_{2009}$

Therefore $y=-F_{2011}$ and $x=F_{\boxed{2012}}$.

See also

2007 Alabama ARML TST (Problems)
Preceded by:
Problem 8 		Followed by:
Problem 10
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15
Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=2007_Alabama_ARML_TST_Problems/Problem_9&oldid=27590"