However, we know that f1=1 which is clearly not divisible by d. Thus, their difference fn+1 - fn = fn-1 will also be divisible by d. Theorem 2 : Consecutive Fibonacci numbers are relativelyprime.Īssume that there exist some two consecutive Fibonacci numbers sayįn and fn+1 that have a common divisor d, where d is greater than 1. Since fmk -1fm is divisible by fm, fmk1fm + 1 is also divisible by fm. Let n be divisible by m, i.e., n = m * k where k is some integer. Theorem 1 : If n is divisible by m, then fn is divisible byfm. Number Theory Properties of the Fibonacci Sequence: = 1 + 1/ lim (fn-1 /fn ) as n approaches infinity = 1 + lim (fn-1 /fn ) as n approaches infinity = lim (1 + fn-1/ fn) as n approaches infinity = lim fn + fn-1 /fn as n approaches infinity = lim f n+1 / f n as n approaches infinity Proof that Rn converges to the Golden Ratio: The sum of the first n Fibonacci numbers with even indices is The sum of the first n Fibonacci numbers with odd indices is Mathematical Properties of the FibonacciSequence :īy adding each of these terms, we get the desired result. The Fibonacci sequence and the Fibonacci numbers also have manyinteresting mathematical properties. the appendages and chambers on many fruits and vegetables such as thelemon, apple, chile, and the artichoke the petals on various flowers such as the cosmo, iris, buttercup, daisy,and the sunflower Some items in nature that are connected to the Fibonaccinumbers are: One of the most fascinating things about the Fibonacci numbers is theirconnection to nature. n = the number of the term, for example, f3 = the thirdFibonacci number and The recursive definition for generating Fibonacci numbers and the Fibonaccisequence is:ġ. For example, the next Fibonaccinumber can be obtained by adding 144 and 89. As can beseen from the Fibonacci sequence, each Fibonacci number is obtained by addingthe two previous Fibonacci numbers together. So, at the end of the year, there will be 144 pairs of rabbits, allresulting from the one original pair born on January 1 of that year.Įach term in the Fibonacci sequence is called a Fibonacci number. How many pairs of rabbits will there be after one year?"įibonacci's Solution: The Fibonacci Sequence!ġ, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144. after reaching the age of two months, each pair produces a mixed pair,(one male, one female), and then another mixed pair each month thereafter and rabbits begin to produce young two months after their own birth Ģ. "Start with a pair of rabbits, (one male and one female) born on January 1.Assume that all months are of equal length and that :ġ. Was one of the first works on equations to be published by a European. Thenow famous problem appeared in Liber abaci which The basic fact from analysis will be some concepts related to power series.Fibonacci numbers and the Fibonacci sequence are prime examplesof "how mathematics is connected to seemingly unrelated things." Even thoughthese numbers were introduced in 1202 in Fibonacci's book Liber abaci,they remain fascinating and mysterious to people today.įibonacci, who was born Leonardo da Pisa "son of Bonaccio", gave a problem inhis book whose solution was the Fibonacci sequence as we know it today. Well there is no specific proof which makes use of calculus (meaning differentiation / integration) however there is a proof which does involve analysis in an indirect way.
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