relation between golden ratio and fibonacci numbers
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The golden ratio is the limit of the ratios of successive terms of the Fibonacci sequence (or any Fibonacci-like sequence), as originally shown by Kepler: Therefore, if a Fibonacci number is divided by its immediate predecessor in the sequence, the quotient approximates
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A method to define the golden ratio is as follows:
Consider a line, split into two parts of unequal length, A and B (where A < B), such that B/A = (A+B)/B. In other words, the ratio of the larger part of the line to the smaller part of the line should be the same as that of the entire line to the larger part of the line.
Now, consider a contiguous subsequence of the Fibonacci sequence A,B,C. We know that C = A+B (according to the definition of the Fibonacci sequence). Now, going by our definition of the line, for the golden ratio, it is achieved if B/A = C/B.
For small numbers, this need not always be true (for instance, 1,1,2 would give 1/1 = 2/1). However, as the position of the numbers increase, the value of the ratio of two consecutive numbers approaches the Golden ratio (approx. 1.618).
1,2,3 -> 2/1->1.5
2,3,5 -> 1.5->1.67
3,5,8 -> 1.67->1.6
5,8,13 -> 1.6->1.625
8,13,21 -> 1.625->1.615
13,21,34 -> 1.615->1.619
21,34,55 -> 1.619->1.6176 ... ... ... (which is very close to the golden ratio)
Hence, as we increase our start position in the Fibonacci sequence, we find that the middle number approaches the golden ratio times the smaller number, and the larger number is the golden ratio times the middle number.
Consider a line, split into two parts of unequal length, A and B (where A < B), such that B/A = (A+B)/B. In other words, the ratio of the larger part of the line to the smaller part of the line should be the same as that of the entire line to the larger part of the line.
Now, consider a contiguous subsequence of the Fibonacci sequence A,B,C. We know that C = A+B (according to the definition of the Fibonacci sequence). Now, going by our definition of the line, for the golden ratio, it is achieved if B/A = C/B.
For small numbers, this need not always be true (for instance, 1,1,2 would give 1/1 = 2/1). However, as the position of the numbers increase, the value of the ratio of two consecutive numbers approaches the Golden ratio (approx. 1.618).
1,2,3 -> 2/1->1.5
2,3,5 -> 1.5->1.67
3,5,8 -> 1.67->1.6
5,8,13 -> 1.6->1.625
8,13,21 -> 1.625->1.615
13,21,34 -> 1.615->1.619
21,34,55 -> 1.619->1.6176 ... ... ... (which is very close to the golden ratio)
Hence, as we increase our start position in the Fibonacci sequence, we find that the middle number approaches the golden ratio times the smaller number, and the larger number is the golden ratio times the middle number.
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