2007 Alabama ARML TST Problems/Problem 9

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 $x=F_{\boxed{2012}}$ .

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

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2007 Alabama ARML TST Problems/Problem 9

Problem

Solution

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