Prove that $F_n$ is a multiple of $F_k$ given that n is a multiple of k, using the identity $F_{a+b+1}=F_{a+1}F_{b+1}+F_aF_b$

Question

I haven't been able to get very far in the problem at all because I can't figure out how to change it to the terms of the identity. All I have for the problem is $F_{kx}$ = $F_n$ and y$F_k$=$F_n$. But I don't understand what this has to do with the identity, where the numbers in the subscript are being added and which does not necessarily have multiples.

Try to use the identity to prove by induction on k that $F_{m}$ divides $F_{mk}$. — Matthew Towers, Nov 05 '19 at 15:26

score 1 · Accepted Answer · answered Nov 05 '19 at 16:07

If $n$ is a multiple of $k$, we can write $n=km$ for some $m \in \mathbb{N}$.

We will prove that $F_n=F_{km}$ is a multiple of $F_k$ by induction on $m$.

$m=1$: then $F_n=F_k$ and clearly $F_n$ is a multiple of $F_k$.

Suppose now that $F_{k(m-1)}$ is a multiple of $F_k$ i.e. $F_{k(m-1)}=c F_k$

Now $$F_{km}=F_{k(m-1)+(k-1)+1}=F_{k(m-1)+1}F_{k-1+1}+F_{k(m-1)}F_{k-1}=F_{k(m-1)+1}F_{k}+cF_{k}F_{k-1}=F_k(F_{k(m-1)+1}+cF_{k-1})$$

therefore $F_n=F_{km}$ is a multiple of $F_k$.

Prove that $F_n$ is a multiple of $F_k$ given that n is a multiple of k, using the identity $F_{a+b+1}=F_{a+1}F_{b+1}+F_aF_b$

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