Fibonacci sequences and the golden section

In the year 1202, the Italian mathematician Leonardo of Pisa, also known as Fibonacci proposed the now famous rabbit problem and showed that the solution is given by a sequence of numbers, Fn defined recursively by Fn+2=Fn+1+Fn,F1 =1F2=1n≥1 .;We explore different topics related to these numbers. We start by discussing generalizations of the Fibonacci Sequence and prove some generalized identities. A general method to obtain series expansion of a given class of functions in terms of the Fibonacci numbers is given. We also find a formula for its coefficients. Some matrix techniques are used to obtain Fibonacci-type identities.;The positive root of the Fibonacci quadratic equation x 2 − x − 1 = 0 is a=1+52 called the golden section. Interestingly enough, it appears frequently in geometrical shapes such as triangles, circles, ellipses, and hyperbolas.