The discovery of the first formula

Current Project
Modified Fibonacci sequence
3-month life-span


The project
How did I obtain the formulas in the previous post?

Fn = Fn-1 + Fn-2
and
Tn = Tn-2 + Tn-3

First, take a look at the data I obtained from Figure 0 and Figure 1.


You can see that the first three terms in both sequences are the same. This makes sense, because the rabbits in the new model don't die until after three months of living.
Let me point out that the formula for the Fibonacci sequence is not something I generated. It has been known for many years, and often the Fibonacci sequence is defined using that formula (instead of rabbit lineage).

Using Fn = Fn-1 + Fn-2 as a guide, I looked at how I could add up two previous terms (and possibly consecutive terms) in order to obtain the current term.

As you can see, we are adding two previous and consecutive terms, but they are not previous to the current term. Rather, they are previous to the previous term. (Yeah, read that all slowly.)

The new question is, how do I prove this formula can be applied to the infinitely many following terms?