I have $f_{n}=f_{n-1}+f_{n-2}; f_{n}= [0,1,1,2,3,5,8,13,21,34,55,89,144,233,...]$ for which I note that indeed, 2 divides 4, and $f_{2}$ divides $f_{4}$.
I am wondering if a proof by induction is sufficient, and exactly how would I go about doing that in this case? Or could I use the closed form formula to prove it: $\dfrac {1} {\sqrt {5}} [( \dfrac {1+\sqrt {5}} {2}) ^{n}-( \dfrac {1-\sqrt {5}} {2}) ^{n}]$
Thanks!
Hint: Use $\dfrac{A^{pq}-B^{pq}}{\sqrt{5}} = \dfrac{(A^p-B^p)}{\sqrt{5}}(A^{p(q-1)}+ A^{p(q-2)}B^p+\cdots + B^{p(q-1)})$. Thus you need to show the second factor is an integer. But you can rewrite it as the sum of the terms $\pm (A^k + B^k)$, and you have to show each $A^k+B^k$ is an integer. And this can be done by induction and is easier than the original one.
you can also use following identity $$f_{kn+c}=\sum_{i=0}^k \dfrac{k!} {i!(k-i)!} \cdot f_{c-i} \cdot f_{n}^i \cdot f_{n+1}^{k-i}$$ which can be easily proved using induction.Plugging in $$c=n$$ your problem becomes trivial
