Fibonacci Numbers

The Fibonacci numbers form a sequence defined recursively by starting with 0 and 1, and then producing the next number by adding the two previous: {0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, ...}. The earliest reference to these types of numbers is contained in a book by Indian mathematician Pingala in 500 BC. It was studied about 1200 AD by Leonardo of Pisa, also known as Fibonacci, to describe the growth of an idealized rabbit population.

Johannes Kepler noticed that the ratio of adjacent Fibonacci numbers converges to the golden mean f (phi). Actually, a sequence can be created by starting with any two real numbers (what about imaginary numbers?) and determining the next number by addition of the previous two. All of these generalized Fibonacci sequences also converge to the golden mean.

ACTIVITY: set up a program in Excel to generate the Fibonacci
sequence; change the initial pair of numbers to show that any
starting pair will still converge to the golden ratio (generalized
Fibonacci sequences)

Phi is an irrational number, approximately 1.618, that possesses many interesting properties. Phi can be approximated on your calculator. Simply take the square root of one, then add one to that answer. Then take the square root of the result, add one again, repeat this over and over until the number remains constant due to the limit of the calculator window width.