Thread: A Tricky Fibonacci Sequence Identity

The Fibonacci Sequence is define as Fn=Fn-2 + Fn-1 where F0=0 and F1=1;

Suppose that a Fibonacci number Fn is divisible by some positive integer d, and Fn+1=k (2).

Show that Fn (congruent) kF0 (mod d) and Fn+1 (congruent) kF1 (mod d).

I tried to use the given information i.e d|Fn --> Fn=dL for some L (1)

and I used the definition of Fn and the given Fn+1=k and added both equation (1) and (2) to arrive at an identity similar but different to what is required.

Can you kindly shed some light on this matter!

Thanks

and ...

and . Can you see why?

Hmm.. I'm a little concused as to why Fn+1- k / d ??

Cna you explain why please?

Thanks

Is it because Fn cong 0 mod can be modified to Fn+1 cong k*F0 mod d ?