Prove Fibonacci formula with induction?

[Sven]

Homework Statement

I'm trying to prove the Fibonacci formula with induction, but I'm having difficulties. This is what I'm trying to prove:

http://www.psc-consulting.ca/fenske/cpjav17e.gif

Homework Equations

The Attempt at a Solution

So I did a base case, n=1. It worked, so move to induction. Now, I have absolutely *no* idea how to get this to work, even start? So I took it and added n+1 to both sides. Then what? I tried multiplying by sqrt(5) to make it a lil simpler (dunno if I'm allowed to do that) but I'm really not sure where to go next. Any help would be very much appreciated.

 

[willem2]

Try to assume that the formula is true for f(n) and for f(n-1).

what base case(s) do you have to prove?

 

[Architeuthis Dux]

I understand your struggle to prove the Fibonacci formula with induction. It can be a challenging task, but with careful reasoning and mathematical manipulation, we can arrive at a proof.

First, let's review the Fibonacci sequence. It is defined as a sequence of numbers where each term is the sum of the two preceding terms, starting with 0 and 1. So, the sequence goes like this: 0, 1, 1, 2, 3, 5, 8, 13, 21, ...

Now, let's rewrite the formula we are trying to prove using the definition of the Fibonacci sequence:

F(n+1) = F(n) + F(n-1)

where F(n) represents the nth term in the Fibonacci sequence.

Next, we will use mathematical induction to prove that this formula holds true for all values of n.

Step 1: Base case
We start by proving the formula for the first term, n=1. From the definition of the Fibonacci sequence, we know that F(1) = 1 and F(0) = 0. Plugging these values into the formula, we get:
F(1+1) = F(1) + F(1-1)
F(2) = F(1) + F(0)
1 = 1 + 0
1 = 1

So, the formula holds true for n=1.

Step 2: Inductive hypothesis
Assuming that the formula holds true for some arbitrary integer k, we will prove that it also holds true for k+1.

Step 3: Inductive step
We want to show that if the formula holds true for k, then it also holds true for k+1. So, we start with the left side of the equation:
F((k+1)+1) = F(k+2)
Using the definition of the Fibonacci sequence, we can rewrite this as:
F(k+2) = F(k+1) + F(k)

Now, we will use our inductive hypothesis and substitute F(k) and F(k+1) with their respective formulas:
F(k+2) = F(k+1) + F(k)
= (F(k) + F(k-1)) + (F(k-1) + F(k-2))
= (F(k) + F(k-1)) + F(k-1) + F(k