Question

Prove, by mathematical induction, that , where is the th Fibonacci number?

Answer 1

To prove the given equation using mathematical induction, we show that it holds true for the base case (n=1) and then assume it holds for a generic case (n=k) and prove it for the next case (n=k+1).

Step-by-step explanation:

To prove the equation using mathematical induction, we need to show that it holds true for the base case (n=1), and then assume it holds for a generic case (n=k) and prove it for the next case (n=k+1).

Base case (n=1):

The first Fibonacci number, F₁, is defined as 1, and the equation holds true.

Assumption (n=k):

Let's assume that the equation holds true for the kth case. So, Fₛ = (1/√5)((1+√5)/2)ˤ - (1/√5)((1-√5)/2)ˤ

Proof (n=k+1):

We need to prove that Fₛ₁ = Fₛ + 1. By substituting the assumption into the equation and simplifying, we can show that the equation holds true for the (k+1) case as well.