How do we use induction to prove the Fibonacci numbers satisfy this formula?

[remaan]

Help with induction !

Homework Statement

Prove that for all n  1, the Fibonacci numbers satisfy
F_n-1 F_n+1 - (F_n)² = (-1)ⁿ:

The Attempt at a Solution

I understand that we have to use induction to prove this.
And we start with a base case of n = 1

But, how do we plug the numbers in the formula above ?
I mean what is F ?
Is there any other givens we should know ?

Am I in the right direction

 

[n.karthick]

F is the Fibonacci number as given below
F0=0
F1=1
F2=1
F3=2
F4=3
F5=5
and so on.

Plug in the values of F for any n and see that the relation given to you holds.

 

[remaan]

Ya - but then how am I going to use induction ?
If I only did what you suggested ?

 

[annoymage]

so, if n=1, it satisfied, assume its true for some n>=1

maybe easiest way to prove from (-1)ⁿ⁺¹=-(-1)ⁿ

and try to show F_n F_n+2 - (F_n+1)²

 

[remaan]

Yes, this is exactly where I got stuck !

I need to show that the formula you've written is the same as the one given in the problem, using the inductive Hypothesis.

To do that : I need to show that each of the corresonding terms are eaqual (to get the over all equality of the eqution)

Fn Fn+1 should be equal Fn-1 Fn+1

and F(n)^2 should be equal to F(n+1)^2


Can you Please give me a hint to continue ?

i was thinking about multiplying the equation by some element/number to get red of the numbers and reach the original one ?

What do you think ?

remaan

 

[annoymage]

hmm, Fibonacci sequence is given by F_a=F_a-1+F_a-2

 

[remaan]

and we can replace the a by n right ?

 

[annoymage]

of course because its true for all a>=2, so -(F_n-1 F_n+1 - (F_n)²) and try to manipulate to get F_n F_n+2 - (F_n+1)²

 

[remaan]

Thanks annoymage -
you're doing a good job making me progress with this -

I will do that and let you know what happens with me

 

[remaan]

But, isn't it the other way around : I meaning shouldn't I start with
Fn Fn+2 - (Fn+1)2

and manipulate it to get : -(Fn-1 Fn+1 - (Fn)2)

Because, this is how standard induction works.

Or, it doesn't really matters ?

 

[annoymage]

its the same thing you want to prove (-1)ⁿ⁺¹=F_n F_n+2 - (F_n+1)²

so if you start with (-1)ⁿ⁺¹, then show (-1)ⁿ⁺¹=-(-1)ⁿ= -(F_n-1 F_n+1 - (F_n)²)=...=F_n F_n+2 - (F_n+1)²

if you start with F_n F_n+2 - (F_n+1)², then show F_n F_n+2 - (F_n+1)²=...=-(F_n-1 F_n+1 - (F_n)²)==-(-1)ⁿ=(-1)ⁿ⁺¹

 

[remaan]

oky Now I see what you mean,
But still working the Algerbra for the manipulation

 

[remaan]

Now, I am trying to use the three equations to get to -(-1)^n+1

However, I not sure that I am in the right direction.

I am trying to replace Fn with its value from the formula you stated, and the same for Fn+2 and Fn+1

Do you think I should continue with that ?

 

[remaan]

Greetings,
So the final thing I got is :
(Fn)^2 + Fn+1 (Fn - Fn+1)

Now how to precede to get what you've mentioned ?

 

[annoymage]

hmm, here's the clue ;P $$F_{n+1}=F_n+F_{n-1} \Rightarrow F_n-F_{n+1}=-F_{n-1}$$

 

[remaan]

Thanks a lot -
I am now done with this problem :)