Proof of Mathematical Induction

[mickadoos]

I have to solve this problem using proof of mathematical induction. Unfortunately, I'm not sure how to go about beginning the body of the proof:

Problem 4:
Theorem: Given the Fibonacci sequence \(\displaystyle \, f_n,\,\) then \(\displaystyle \, f_{n+2}^2\, -\, f_{n+1}^2\, =\, f_n\, f_{n+3},\, \forall n\, \in\, \mathbb{N}\)

Suppose I've proved the theorem true up to a certain value k. Then considering k+1, f²_K+3-f_k+2²=f_K+1*f_K+4.

Now how would I prove the theorem for K+1, given that I've proved it for k?

 

[pka]

I have to solve this problem using proof of mathematical induction. Unfortunately, I'm not sure how to go about beginning the body of the proof:

Problem 4:
Theorem: Given the Fibonacci sequence \(\displaystyle \, f_n,\,\) then \(\displaystyle \, f_{n+2}^2\, -\, f_{n+1}^2\, =\, f_n\, f_{n+3},\, \forall n\, \in\, \mathbb{N}\)

Suppose I've proved the theorem true up to a certain value k. Then considering k+1, f²_K+3-f_k+2²=f_K+1*f_K+4.

Now how would I prove the theorem for K+1, given that I've proved it for k?

First you need to use the definition: for \(\displaystyle n\ge 3\) we have \(\displaystyle f_n=f_{n-1}+f_{n-2}\).

So \(\displaystyle f_{ n+3}^2=(f_{n+2}+f_{n+1})^2=f_{n+2}^2+2f_{n+2}\cdotf_{n+1}+f_{n+1}^2\)

From there on it becomes a game of subscripts. Good luck!