For this final assignment we will be exploring the beautiful Fibonnaci Sequence. Pronounced: FIB-OH-NOT-CHYEE. Before we begin our explorations with varying n and ratios let's review the characteristics and properties of the Fibonnaci Sequence. The Fibonnaci Sequence was named after the Italian mathematician, Leonard Pisano Bigollo. The sequence is found by adding the two numbers before it, where f(0) = 1 and f(1) = 1. The sequence is shown in the tables below with the function f(n) in column B. The Fibonnaci Sequence is found throughout nature and of course mathematics. Observe the pictures below, but before continuing please make sure your browser is maximized so all information and data will be seen.

 

 

Leonard Pisano Bigollo 'Fibonnaci'

 

So what is so special about the Fibonnaci Sequence? Why is it associated with the spiraling figures above? The Fibonnaci Sequence is particularly special because as n increases our ratio of two successive entries, f(n + 1)/f(n), gets closer and closer to the Golden Ratio! It takes a few entries to get a ratio approximately equivalent to the golden ratio.

 

Let's show mathematically as n increases the ratio f(n + 1)/f(n) becomes the Golden Ratio.

We say that the function f(n + 1) and f(n) are in the golden ratio if the proportion b : a is the same as the analogous proportion a : (b - a) in the smaller triangle, i.e. if f(n + 1)/f(n) = f(n)/ (f(n + 1) - f(n)) then f(n + 1) and f(n) are in the golden proportion if and only if f(n + 1)= f(n)*((1 + √5)/2)). For simplicity let's let b = f(n + 1) and let a = f(n).

 

We have proven the ratio f(n + 1)/ f(n) does in fact become the Golden Ratio, and we can see that f(n + 2)/ f(n) is equivalent to the Golden Ratio plus one!

 

 

If we let f(1) = 3 and f(1) = 4 we see that our ratios are consistent with the Golden Ratio and the Golden Ratio plus one.

 

In conclusion, we see that our ratios f(n + 1)/ f(n) converge to the Golden Ratio or the Golden Ratio plus one when heading towards infinity, and even when f(1) = 3 or 4 our ratios remain Golden or Golden plus one. Thanks Fibonnaci!