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The Fibonacci numbers \(F_a, F_b, F_c\) form an increasing arithmetic sequence.  If a+b+c = 2000, compute a.

 

Thanks if you answer, this will help a lot! :)

 May 28, 2021
 #1
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I used a computer program, and found a = 624.

 May 28, 2021
 #2
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After typing in the answer, the website said it was incorrect.  If you gave the method, it would be greatly appreciated though.  

Guest May 28, 2021
 #3
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Never mind, after experimenting a bit, I figured out the answer.  

 

Essentially, I started by assuming \(F_c=F_{b+1}.\)  Thus, \( F_c-F_b = F_b+F_{b-1}-F_{b} = F_{b-1}\)

 

Knowing that Fa, Fb, and Fc form an arithmetic sequence, we can use properties of Fibonacci numbers to write Fa, Fb, and Fc in terms of

Fb, and use the fact that a+b+c=2000 to find what one of them is.  The rest is simple.  

 

Fa=F665, a=b-2, c=b+1

 

I know I used LaTeX at the start, and it looks clear, but it was too hard to write quickly.  

Guest May 28, 2021

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