Fibonacci Numbers Proof by Induction (Looking for Feedback)

Question

I recently typed a short proof for a homework problem regarding Fibonacci numbers. For some background, I am a physicist and not a mathematician so I was hoping to share my proof with some mathematicians for two reasons. I am new to typing formal proofs, so I am more interested in knowing if the wording/syntax is properly done. I'd also like to know if the formatting is acceptable (I typed this in Latex, something else I am new to doing). To be clear, I'm not looking for help on the problem itself since this is homework. I just want to know if this proof would be acceptable in the mathematics community for my own personal benefit.

Also, just assume you know this problem is about Fibonacci numbers, even though I didn't state it explicitly in the proof.

Picture of my Proof

Answer 1

The proof is technically valid and well-written; however, you never use the inductive hypothesis in your inductive step. Thus invoking induction was unnecessary.

Answer 2

Your proof is more of a direct proof rather than a proof by induction. You didn't really utilize the induction hypothesis in the actual induction step. Nowhere did you use the fact that $F_{k-2}+F_{k+2}=3F_k$. That's not to say that your proof is incorrect, it just doesn't necessarily do it in the way requested. This brings up the question of the semantics of "what makes a proof specifically an induction proof or not an induction proof?" I for one am in the camp of an induction proof specifically makes use of an implication of the form $P(n)\implies P(n+1)$.

As for how to organize an induction proof, this post has some good advice.

As for how to use the induction hypothesis in your actual problem, try to rewrite $F_{k-1}+F_{k+3}$ as sums of fibonacci numbers which are four apart with slightly smaller indices.

(You might find it convenient to have your induction hypothesis instead be the statement $P(k)\equiv(F_{k-3}+F_{k+1}=3F_{k-1})\wedge(F_{k-2}+F_{k+2}=3F_k)$ which would require an additional base case, or you could phrase this instead as strong induction)

$F_{k-1}+F_{k+3}=F_{k-3}+F_{k-2}+F_{k+1}+F_{k+2}$ by Fibonacci recurrence

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$=(F_{k-3}+F_{k+1})+(F_{k-2}+F_{k+2})$ by rearranging

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$3F_{k-1}+3F_k$ by induction hypothesis...

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$3(F_{k-1}+F_k)=3F_{k+1}$ by factoring and Fibonacci recurrence.