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.
(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,
and we want show
Notice that we can rewrite this last equation as
The sum on the righthand side of this is our induction hypothesis, so we have
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.