The Fibonacci Numbers and Golden Ratio II

_{September 13, 2021.}

Continuing from the previous post, the only thing left to do now is to find a formula for Fibonacci from the information obtained. We all know that a Fibonacci number is represented in the form of its two previous numbers, that is, F_{n} = F_{n-1} + F_{n-2}. So, now the question is can't we find it without its previous numbers? We can! It is called an explicit formula. Let's get started with it.

In the last post, we talked about a statement that ɸ^{n} = F_{n}ɸ+F_{n-1}. Here, we have to understand that this statement holds for ɸ̄ too.

Therefore, we can say ɸ̄^{n} = F_{n}ɸ̄+F_{n-1}.

Let's just subtract one from another,

ɸ^{n} - ɸ̄^{n} = F_{n}ɸ+F_{n-1}-(F_{n}ɸ̄+F_{n-1})

= F_{n}(ɸ-ɸ̄) = F

From this we can say, F_{n} = ɸ^{n}-ɸ̄^{n}/√5

What this means? Well, ɸ^{n}/√5 is a little smaller or larger than F_{n}, and subtracting off the ɸ̄^{n}/√5 gives us the exact F_{n}.

So, now the formula for computing the Fibonacci numbers without the computation of it's previous values can be given as:

F_{n} = [ɸ^{n}/√5]

Well, this is super cool! There is a lot of interesting stuff to do with Fibonacci. Maybe I will try to do those if I get the interest.