Understanding the recursive relationship that the Fibonacci sequences uses is rather simple.
Starting with F0=F1=1 and n>1, it can be seen how the recursive formula
Fn=Fn-1+Fn-2
leads to the sequence as we normally see it: 1, 1, 2, 3, 5, 8, 13, ... This recursive formula can be written in a closed form, meaning the nth Fibonacci number can be calculated using this formula.
It is known as the Binet Formula and is expressed as:
(Bortner & Peterson)
or 
(Supriatna, Carnia, & Ndii).
Some people dispute that Binet was not the first to discover this formula, but that it was found by de Moivre in the 18th century (Supriatna, Carnia, & Ndii).
Something interesting about the Golden Ratio is that it is equivalent to its own reciprocal with the addition of one: 
(Omotehinwa & Ramon, p. 631).
One of the most prominent applications of the Golden Ratio is in the creation of golden sections.
Golden sections are created when a line segment is divided into two unequal parts and the length of the longer part divided by the length of the shorter part is equivalent to the length of the entire line segment divided by the length of longer part.
In symbolic terms, 
where a is the length of the longer part of the line segment and b is the length of the shorter part of the line segment (Kazlacheva).
Another way to represent this is by looking at a line segment with length 1 and the length of the longer part equal to x with the length of the shorter part as 1-x.
This leads to the ratio of 
This can be manipulated to become the quadratic equation x2+x-1=0.
When we use the quadratic formula to find the roots of this polynomial, we find that 
As we look at the positive root, it can be seen that 
and
(Omotehinwa & Ramon, p. 631).
An interesting connection between the Fibonacci sequence and other mathematics is in Pascal's Triangle.
Pascal's Triangle is a visual representation of finding the number of combinations of choosing k items from n total items where 0 ≤ k ≤ n.
Each row in Pascal's Triangle represents the n and begins with n=0 and the number each column is the k spot.
If we go to the 5th row where n=4 we can see that "4 choose 0" or 
is equal to 1.
As the row continues, it shows 
When Pascal's Triangle is left aligned and the diagonals are summed together, they are equivalent to the numbers in the Fibonacci sequence, as show in the picture (Reich).
Another interesting feature of the Golden Ratio is that it can be written as the expression of "an infinite cascade of square roots" and "an infinite cascade of fractions" (Reich).
These derivations follow:
and 