SIR Mathematics Investigation Abstract
THE TRUTH ABOUT FIBONACCI: NUMBER PROPERTIES YOU NEVER THOUGHT POSSIBLE!
Presenter:
Harrison S. Stein, Illinois Mathematics and Science Academy, 1500 West Sullivan Road, Aurora, IL, 60506; harrsman@imsa.edu
Advisor:
Dr. Steven Condie, Illinois Mathematics and Science Academy, 1500 Sullivan Road , Mathematics, Aurora, IL, 60506; 630-907-5967; scondie@imsa.edu
Abstract:
It is well known that in a Fibonacci sequence, the square of any term minus the product of the two surrounding terms alternates between 1 and -1 depending on which terms are calculated. It is not nearly as well known, however, that when the same operation is performed on a Fibonacci-type series (a series, Qn, with Q1= 1, Q2= x and Qn= Q n-1 - Q n-2) the results alternates between +/- (x2 - x - 1 ) and thus, depends only on x. Amazingly, the roots of x2 - x - 1 = 0 are (1 +/- square root of 5) / 2, the positive of which yields the golden ratio! I subsequently proved a s general case of the same property involving a Fibonacci-type sequence such that Q1=y and Q2 = x. Instead of subtracting the product of the surrounding terms, I subtracted the product of the two terms that are distance m away from our central term. This difference also happens to alternate, this time between +/- [(fm)2 * (x2 - xy - y2)], where fm is the mth Fibonacci number. This is a novel discovery of a very unique property of Fibonacci numbers. In order to further understand the sequence, I am currently expanding my research, notably with a two-dimensional array of Fibonacci sequences. Eventually, I hope to generalize my results for an N-dimensional array of sequences.