Should have a strong background in Mathematics . Applications of Fibonacci numbers

Must be able to write in LaTex using an Overleaf account and provide the error free code once the document is finished. I must be able to verify the code works properly with no errors in overleaf. I am providing 10 references to be used to write the research paper may add additional sources if necessary. Sources should not be from Wikipedia, should come from journals, online libraries scientific studies, etc. Sources must be correctly cited when used. I will attach a basic outline of the materials that should be covered. The first portion of the paper will include the basic properties of the Fibonacci sequence. Define closed-form and recursive form. Relation to the golden ratio must state and prove. Relation to Lucas sequences state and prove. Viewed as a linear recurrence equation with a sample computation. State and prove Binet’s formula, Cassini’s identity. State and prove generating function, the sum of Fibonacci numbers and sum of Fibonacci numbers squared. When proving the definition must include written steps that explain what computations or manipulations that were performed to get proof . Must include all proper mathematical proofs and symbols that accompany the proof. The second part will include applications of Fibonacci. Where can the Fibonacci sequence be found, where was it used in the past and present. This will show how Fibonacci is applied in each one of these categories. Geometry and Art examples spirals, golden rectangle and golden angle. Fibonacci was found to present in Michael Angelo’s painting, sculptures pyramids, etc. Where is Fibonacci sequence found in Music examples of Fibonacci in 20th-century music the shape of the violin and piano keys, etc. Nature leaf patterns, honey bee population flowers exhibiting Fibonacci petal sequence. Trading examples in Fibonacci retracement etc. Anatomy and physiology examples of how Fibonacci is seen in the hand, human DNA, etc. Physics examples of how Fibonacci sequence is in chaos, hydrogen bonds. Chemistry examples of Fibonacci sequence in quasicrystals. Astrophysics examples of Fibonacci in black holes. Technology examples of Fibonacci in quantum computing. please use the examples provided may use more examples as long as they are cited with a valid reference. I do not need an abstract, nor do I need any history on the life of Fibonacci. I am only looking for computations and applications of the sequence.

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