Fibonnaci Sequence

By

Audrey V.
Simmons

The Fibonnaci Sequence is a most amazing sequence of numbers
that appears in different problems and is related to the golden ratio of or
1.618034. We will look at how
these two prevalent items come together.

# PART A: Calculate the
Fibonnaci Sequence/Determine Ratios

The Fibonnaci Sequence is a sequence of numbers that is
calculated by adding the previous two numbers to get the next value. For example,

F(0) = 1, f(1)=1, f(2) = 2, f(3) = 3, f(4) = 5,…f(n) = f(n-1) + f(n-2).

These numbers are represented in column A. To find the numbers in column C (B was
skipped to make the numbers more readable), take the ratio of adjacent terms.
A3/A2 = 2, A4/A3 = 1.5, A5/A4 = 1.66666667, and so on.

The ratio of the numbers becomes the golden ratio.

Let’s try the ratio of every second term to determine
if there is any relationship.

The ratio of every second term is the golden number plus
one.

Would the next ratio be the golden ratio plus two?

If we look at column G which represents the ratio of every
third term, we can see that 4.23606798 is not the golden ratio plus two. However, it is the golden ratio times
two plus one.

In column I when we find the ratio of every fourth term the
answer approaches the number 6.85410197. The golden ratio times three plus two.

Column K(ratio of every fifth term), the number 11.0901699
is the golden ratio times 5 plus three.

Below is a graph of the respective ratios showing the
constant nature of the amounts.

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## PART B: Arbitrary
numbers and their ratios

This time random numbers have been picked to begin the
sequence, but we have used the same formula f(n+1) = f(n) +f(n-1) to calculate our numbers.

Even if the initial numbers are different, when the
Fibonnaci formula is used to calculate the terms the respective ratios reach
the same values as the values in the Fibonnaci ratios.

RETURN