Daily 7: A Few Fibonacci Propositions

After hunting around the internet, I came across several propositions made by Fibonacci and theorems regarding the Fibonacci sequence. Here are proofs of a couple of the things I came across. (Knowledge of HTML coding is lacking, so some of the equations look pretty bad, but it should be somewhat clear what I mean.)

First we prove a proposition made by Fibonacci that states the difference between two squared consecutive numbers is the sum of the two numbers. For example, 5² - 4² = 9 = 5 + 4. The proof is pretty simple, but it's good for a warmup.

Let n be any integer. We will prove that (n + 1)² - n² = (n + 1) + n directly. Distributing the lefthand side of this equation, we get

(n + 1)² - n² = n² + 2n + 1 - n².

Simplifying gives

(n + 1)² - n² = 2n + 1 = (n + 1) + n,

which was to be shown. So, the difference between two squared consecutive numbers is the sum of the two numbers. Next, we prove a formula for the sum of the Fibonacci numbers from 1 to n with an inductive proof.

Using wikipedia, I found that the sum of all Fibonacci numbers up to n is one less than the (n + 2)th term. That is, if F(n) is the nth Fibonacci number,

∑k =1nFk=   F   n+2  −1.

For the basis step, notice that F(1) + F(2) = 1 + 1 = 2 = 3 - 1 = F(4) - 1. With the basis for induction established, we now assume for the inductive hypothesis that

∑i =1kFi=   F   k+2 −1

and we want show

∑i=1k+1Fi=   F   k+3 −1.

Notice that we can rewrite this last equation as

∑i=1k+1Fi=   ∑     i=1     k  F  i+Fk+1.   

The sum on the righthand side of this is our induction hypothesis, so we have

∑i=1k+1Fi=   F   k+2 −1 + Fk+1.

We know that Fk + 1 + Fk + 2 = Fk + 3. Thus, the previous equation can be written

∑i=1k+1Fi=   F   k+3 −1,

which completes the proof. Thus, we have shown that the sum of the Fibonacci numbers to the nth term is one less than the (n + 2)th term. Many of these inductive proofs involving sums seem to be similar, and having proved the sum of natural numbers helped in my understanding of this proof. I remember doing many proofs like this in 210, and I always thought the many patterns that came about from this sequence were interesting. My plan is to continue working with the Fibonacci sequence in my next weekly entry.