I am trying to make a conjecture as to Fibonacci numbers which are divisible by 3 and trying to prove it by mathematical induction where the initial conditons are 0 and 1.
My problem is that I handled the basis step of induction but i was not able to write inductive step.Can you help me, please?
By the way, I found that it is divisible by 3 where F0=0 , F4=3 , F8=21 ,F12=144 ... are divisible by 3, and F0=0 is divisible by 3 is basis step.
I think you are trying to say $F_{4k}$ are divisible by 3 for all $k\geq 0$ .
For the inductive step $F_{4k}= F_{4k-1} + F_{4k-2} = 2 F_{4k-2}+ F_{4k-3}= 3 F_{4k-3} + 2 F_{4k-4}$.
I think you can conclude from here.