Prove that if Fn is the "nth" Fibonacci number that if 3|n (3 divides n) then 2|Fn
(2 divides the nth Fibonacci number).
THIS PROBLEM HAS BEEN SOLVED.
THANKS TO ALL WHO HELPED!!
Prove that if Fn is the "nth" Fibonacci number that if 3|n (3 divides n) then 2|Fn
(2 divides the nth Fibonacci number).
THIS PROBLEM HAS BEEN SOLVED.
THANKS TO ALL WHO HELPED!!
In…
http://mathworld.wolfram.com/FibonacciNumber.html
… you can find the following ‘finite sums’…
(1)
(2)
Since , it follows from (2) that devides …
Regards
Read this post here, it is even more general than what you need. (It uses the Euclidean Algorithm)
Proofs for the formulas that appear above in this thread:
See here
Simply consider . We have , the rest follows like in the previous link.
(Again )
In fact this gets even more general since we have: (this may be proven by induction quite easily since
) thus:
And it follows that we have:
From there check that is a multiple of