Fibonacci sequence divisibility proof

Hello All,

I am thoroughly lost with how to construct this proof:

Let F(sub)n be the nth Fibonacci number.

Prove that, if 3|n then 2|F(sub)n.

Would induction be a recommendation, and if so, could anyone please recommend a base assumption to begin with?

Thank you very much for your time,

Panglot

Quote:

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Originally Posted by panglot

Hello All,

I am thoroughly lost with how to construct this proof:

Let F(sub)n be the nth Fibonacci number.

Prove that, if 3|n then 2|F(sub)n.

Would induction be a recommendation, and if so, could anyone please recommend a base assumption to begin with?

Thank you very much for your time,

Panglot

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If and only if can be placed.


Are you familiar with the theorem that says:

if and only if .

What you want to prove is that

is divisible by 2 for any positive integer n. It is easy to do that by induction on n.

If n= 1,

which is divisible by 2.

Assume that, for some k,

is divisible by 2. That is, that

for some integer m.

Then
.

Quote:

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Originally Posted by HallsofIvy

What you want to prove is that

is divisible by 2 for any positive integer n. It is easy to do that by induction on n.

If n= 1,

which is divisible by 2.

Assume that, for some k,

is divisible by 2. That is, that

for some integer m.

Then
.

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I'm sorry, but I don't quite follow how you arrived at the last two in the series of equivalences:

= F_{3k+1}+ (F_{3k}+ F_{3k+1})= 2m+ 2F_{3k+1}

Quote:

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Originally Posted by panglot

I'm sorry, but I don't quite follow how you arrived at the last two in the series of equivalences:

= F_{3k+1}+ (F_{3k}+ F_{3k+1})= 2m+ 2F_{3k+1}

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By definition of Fibonacci sequence.

F_{k+2}=F_{k+1}+F_{k}.