Applications Involving the Fibonacci Sequence
We can use the Fibonacci sequence and apply the numbers within other applications seen within mathematics. In particular, we can use Fibonacci numbers to find the greatest common divisor by utilizing another mathematical application in number theory, the Euclidean Algorithm. In addition, the Fibonacci sequence is used for another application when it comes to conversions between miles and kilometers. 
To demonstrate this idea, let's look at an example to find the \(\gcd\) of \(f_{11}\) and \(f_{12.}\) From Binet's Formula above, we can calculate and see that \(f_{10}=89\) and \(f_{12}=144.\) We can compute the Euclidean Algorithm by repeatedly dividing till we end up with our last nonzero remainder.