Proving the Fibonacci Sequence Using Induction

[kathrynag]

Homework Statement

F_{1}+F_{3}+F_{2n-1}=F_{2n}

Homework Equations

The Attempt at a Solution

P(k+1):F_{2k-1}+F_{2k+1}=F_{2k+2}

 

[mutton]

What happened to F_1 and F_3?

 

[kathrynag]

What happened to F_1 and F_3?

well, it's F_1+F_3+...+F_2n-1

 

[mutton]

Then P(k + 1) needs to be changed accordingly.

What have you tried so far?

 

[kathrynag]

I've tried plugging in numbers.

 

[mutton]

And how did that work out?

The definition of Fibonacci numbers will be helpful in the induction proof.

 

[kathrynag]

If I plug in 1, I just get F_1, so 1=1
If I plug in 2, I get F_3, so 1+2=F_4, 3=3

 

[kathrynag]

well, it's F_1+F_3+...+F_2n-1

F-1+F-3+...+F_2k-1+F_2k+1
P(k)+F_2k+1
F_2k+F_2k+1

Now I'm stumped...

 

[HallsofIvy]

Homework Statement

F_{1}+F_{3}+F_{2n-1}=F_{2n}

Homework Equations

The Attempt at a Solution

P(k+1):F_{2k-1}+F_{2k+1}=F_{2k+2}

well, it's F_1+F_3+...+F_2n-1

What was "F_1+ F_3+ ...+ F_2n-1"?

In your first post you said the problem was to prove that
F_{1}+F_{3}+F_{2n-1}=F_{2n}

Are you saying now it is actually to prove that
F_{1}+F_{3}+\cdot\cdot\cdot +F_{2n-1}=F_{2n}?

 

[mutton]

F-1+F-3+...+F_2k-1+F_2k+1
P(k)+F_2k+1
F_2k+F_2k+1

Now I'm stumped...

Very close. What happens when 2 consecutive Fibonacci numbers are added?

 

[kathrynag]

Very close. What happens when 2 consecutive Fibonacci numbers are added?

It equals the 3rd Fibonnacci number.
F_1+F-2=F_3

so F_2k+F_2k+1=F_2k+1+1

 

[mutton]

And that's exactly what you wanted to show.

 

[kathrynag]

Ok thanks!

 

[epenguin]

Sorry but maybe the problem is not properly stated? Are the F defined to be Fibonacci numbers?

Then F_2n = F_2n-1 + F_2n-2

So if you are then asking also that

F_2n = F_2n-1 + F₁ + F₃

then

F_2n-2 = F₁ + F₃

which is not making much sense.