Using normal properties of Fibonacci Numbers Prove that:

(F(n+1)^2)+(F(n)^2)= F(2n+1).

We see that F(3)=2 and F(4)=3 and so F(7)=13 so it is true

Not sure where to start here and any help would be appreciated.

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- Mar 2nd 2008, 11:04 AMMatsSundinFibonacci Sequence property
Using normal properties of Fibonacci Numbers Prove that:

(F(n+1)^2)+(F(n)^2)= F(2n+1).

We see that F(3)=2 and F(4)=3 and so F(7)=13 so it is true

Not sure where to start here and any help would be appreciated. - Mar 2nd 2008, 11:36 AMJhevon
- Mar 2nd 2008, 11:58 AMgalactus
I proved this identity back in school using Binet's formula(without induction) and some identities.

Do you have to use induction?. That is a good way to go, but I like this method I stumbled onto a while back while studying Fibonacci's for a class called Seminars in Mathematics. I doubt if it's original, but it works.

I will use

Take note that and and

We will use these later.

From Binet's formula:

We can sub it in our formula to prove:

Expand:

Group and factor:

Take note that everything inside the brackets is 0, due to the identities previously mentioned.

We get:

,

You can show various identities using these tools. They can be handy to know. - Mar 2nd 2008, 05:29 PMMatsSundin
No induction is not needed, it can be proven any way possible. Your solution is quite intuitive and impressive thank you!