Interesting points to notice about Fibonacci Numbers
I. The series begins with 1 and each successive number is the sum of the two previous numbers.
II. The series is made up of even, odd, and prime numbers.
(a) Primes {2, 3, 5, 13, 89, 233, 1597, 28657, ... }
(b) Even {8, 34, 144, 610, 2584, ... }
Except for the primes, all Fibonacci Numbers can be factored.
III. Except for the first few numbers, (a) any given number in the sequence is about 1.618 times the proceeding number, and (b) about 0.618 times the following number.
IV. Looking at the values that are next to each other and dividing by their sums, we see the following:
(1, 2, 3) 1/3 = 0.333 2/3 = 0.666
(2, 3, 5) 2/5 = 0.400 3/5 = 0.600
(3, 5, 8) 3/8 = 0.375 5/8 = 0.625
(5, 8, 13) 5/13 = 0.385 8/13 = 0.615
(8, 13, 21) 8/21 = 0.381 13/21 = 0.619
(13, 21, 34) 13/34 = 0.382 21/34 = 0.618
(21, 34, 55) 21/55 = 0.382 34/55 = 0.618
If more decimal places were used, one would see the interesting fact that the numbers begin to converge to 0.382 (38.2%) and 0.618 (61.8%).
V. Other Fibonacci percents can be determined by dividing numbers in the sequence above them. For example:
(a) Divide the first number by the second. This is 1/1, which is 100%.
(b) Divide the second number in the sequence by the third. This is 1/2 or 0.50, which is 50%.
(c) Taking numbers that are not at the beginning of the sequence, as illustrated in III above, divide the number by the one following it, and the numbers converge to 61.8%.
(d) Taking numbers that are not at the beginning of the sequence, as illustrated in III above, divide a number by two numbers above it, and the numbers converge to 38.2%.
(e) Taking numbers that are not at the beginning of the sequence and divide a number by three numbers above it, the numbers converge to 23.6%.
(f) If one calculates the value of 1 minus each of these Fibonacci calculations, one obtains the values 38.2%, 50%, 61.8%, and 76.4%.
The upshot are the Fibonacci percents:
{0%, 23.6%, 38.2%, 50%, 61.8%, 76.4%, 100%}.
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