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Two sequences \(A=\{a_0, a_1, a_2,\ldots\}\) and \(B=\{b_0,b_1,b_2,\ldots\}\) are defined as follows:

 

\(a_0=0, ~a_1=1, ~a_n= a_{n-1} +b_{n-2} \hspace{2mm}\text{for}\hspace{2mm} n\ge2\)

 

\(b_0=1, ~b_1=2, ~b_n=a_{n-2} +b_{n-1}\hspace{2mm}\text{for}\hspace{2mm} n\ge2\)

 

What is the remainder when \(a_{50}+b_{50}\) is divided by \(5\)?

 

 

 

Hey, I took the time to write this in LaTeX so yall can read it.

 Nov 2, 2020
 #1
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Using a computer program, a_{50} leaves a remainder of 1, and b_{50} leaves a remainder of 2, so a_{50} + b_{50} leaves a remainder of 3.

 Nov 2, 2020
 #2
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a(50) = It is simply the 50th Fibonacci number =12586269025

b(50) = It is the 52nd Fibonacci number             =32951280099

 

a(50)  +  b(50) =[12586269025 + 32951280099] mod 5 == 4

 Nov 2, 2020
 #3
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Have you written out the beginning of the sequence of yourself?

 

What did you do yourself before you asked for help?

 Nov 2, 2020

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