Fibonacci Sequence question

Hey all,

Im hoping you guys could point me in the right direction after getting myself rather lost with a question on my assignment. I am not putting the actual question as i would like to work it out myself. here is a question i have made up that is similar with what i think to be correct. Could anyone verify this or point me in the right direction?

Thanks!

Let Fn be the Fibonacci sequence.

Use the Fibonacci recurrence relation to express Fn+3 in terms of Fn+1 and Fn. Hence show that Fn+1 = 1/2(Fn+5 - Fn) for n=0,1,2,...

the next question i have is the same but with Fn+1 = 1/2(Fn+5 + Fn)

then i have to decide wether the formulas would remain true if the sequence Fn were replaced by a sequence with the same recurrence relation as the Fibonacci sequence but with different initial terms, then justify.

So if i look just at the 1st one at the moment, i think that i would get something like this;

Fn+1 = 1/2(Fn+5 - Fn) for n=0,1,2,...

Fn = 1/2(Fn+1 - Fn+1)
Fn = 1/2(Fn+2 + Fn+1)

if that is not correct, how do i work this out properly as im rather stumped :-S

Quote:

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Originally Posted by Arkious

Hence show that Fn+1 = 1/2(Fn+5 - Fn) for n=0,1,2,...

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but this is not true.

Presumably you meant $\displaystyle F_{n+1}=\frac{1}{2}(F_{n+3}-F_n)$.

Yeah that's the question I actually have, I just didn't want to ask the question I have been given. I'm not 100% with Fibonacci yet. But if you could explain how I would complete the question you have mentioned, I would much much appreciate it

Dont know what you have asked above, the notation is confusing, It gives 2 different expressions for Fn+1 alone. I will try to give you the proof for Fn+1 = 1/2(Fn+3 - Fn).

Fn+3 = Fn+2 + Fn+1
= (Fn+1 + Fn) + Fn+1
= 2Fn+1 + Fn.

Just rearrange terms here.

Salahuddin
Maths online

Yeah that's the problem, it's very confusing!

If i didn't have to show that Fn+1 = 1/2(Fn+3 - Fn) for n=0,1,2,... I would have been able to do it.

Thanks for your input :-) anyone got any ideas though? I'm no further forward.

Did you try to express $\displaystyle F_{n+3}$ in terms of $\displaystyle F_{n+1}$ and $\displaystyle F_n$ ?

You could start by writing $\displaystyle F_{n+3}$ in terms of $\displaystyle F_{n+2}$ and $\displaystyle F_{n+1}$.

firstly thanks for your replies,

right, this is what i have done so far...

in Fibonacci terms;
Fn+3 = Fn+2 + Fn+1
Fn+2 = (Fn+2 + Fn)
Fn+1 = (Fn+2 - Fn)
Fn = (Fn+2 -Fn+1)

(if thats correct)

But then i need to express Fn+1 = 1/2(Fn+3 - Fn)

in Fibonacci terms;
Fn+3 = Fn+2 + Fn+1
Fn+2 = (Fn+1 + Fn)

Correct for these two, but now, get rid of the Fn+2 in the first equation, since we do not have any Fn+2 in our formula. Substitute your Fn+2 = Fn + Fn+1 there, and rearrange terms.

Salahuddin
Maths online

but this still doesnt make sense... how do i then show that Fn+1 = 1/2(Fn+3 - Fn).

surely above is all wrong?

Fn+3 = Fn+2 + Fn+1
(2) = (1) + (1)
Fn+2 = Fn+1 + Fn
(1) = (1) + (0)
Fn+1 = Fn - Fn-1
(1) = (0) - (-1)
Fn = Fn-1 - Fn-2
(0) = (-1) - (-1)

Thats proved that those fibonacci terms are correct, but then how do i then show Fn+1 = 1/2(Fn+3 - Fn)?

Guys, here is the steps.

Fn+3 = Fn+2 + Fn+1
But since Fn+2 = Fn+1 + Fn

Fn+3 = Fn+2 + Fn+1
= (Fn+1 + Fn) + Fn+1
= 2Fn+1 + Fn

Remove Fn, but subtracting it both sides,

Fn+3 - Fn = 2Fn+1.

In other words,

Fn+1 = 1/2(Fn+3 - Fn). This is the same style for many other problems, get it?

Salahuddin
Maths online

Fn+1 = 1/2(Fn+3 - Fn)
(5) = 1/2 ((13) - (3))
(5) = 1/2 (10)

am i supposed to find the solution for Fn then use that to find Fn+3?

I have tried multiple ways and cant find a solution that works for more than one number :-(

Hi Arkious, could you be more specific, I am not able to understand what you are saying. (Basically if you have Fn and Fn+1, you can find Fn+3. But why are you finding the solutions, what is the actual problem?).

Salahuddin
Maths online

Let Fn be the Fibonacci sequence.

Use the Fibonacci recurrence relation to express Fn+3 in terms of Fn+1 and Fn. Hence show that Fn+1 = 1/2(Fn+3 - Fn) for n=0,1,2,...

That is the question i have to answer. My issue is how to show that Fn+1 = 1/2(Fn+3 - Fn).

I dont think that these are of any use for the question i have to answer.

Fn+3 = Fn+2 + Fn+1
Fn+2 = Fn+1 + Fn
Fn+1 = Fn - Fn-1
Fn = Fn-1 - Fn-2

All i need to know is how to answer that question and i think that what i have done so far has confused me more. I just don't understand how im supposed to show that Fn+1 = 1/2(Fn+3 - Fn) for Fn+3 & Fn+1

Quote:

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Originally Posted by Arkious

Fn+3 = Fn+2 + Fn+1
Fn+2 = Fn+1 + Fn

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These are what you need.

Use the second one to substitute for Fn+2 in the first.

Fn+3 = Fn+2 + Fn+1

becomes

Fn+3 = _?_+_?_ + Fn+1.