The ratio of dividing the larger Fibonacci number into the smaller Fibonacci number gives you the golden ratio (1.618 to 1).
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The Golden Ratio is the number (1+sqrt(5))/2~=1.618
The Fibonacci sequence is 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, ... .
Skipping the first two terms, if you divide one term in this sequence by the previous term the resulting sequence converges to the Golden Ratio:
1.0000
2.0000
1.5000
1.6667
1.6000
1.6250
1.6154
1.6190
1.6176
1.6182
1.6180
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The ratio of successive terms in the Fibonacci sequence approaches the Golden ratio as the number of terms increases.
As you expand the Fibonacci series, each new value in proportion to the previous approaches the Golden Ratio.
The "golden ratio" is the limit of the ratio between consecutive terms of the Fibonacci series. That means that when you take two consecutive terms out of your Fibonacci series and divide them, the quotient is near the golden ratio, and the longer the piece of the Fibonacci series is that you use, the nearer the quotient is. The Fibonacci series has the property that it converges quickly, so even if you only look at the quotient of, say, the 9th and 10th terms, you're already going to be darn close. The exact value of the golden ratio is [1 + sqrt(5)]/2
The Fibonacci sequence can be used to determine the golden ratio. If you divide a term in the sequence by its predecessor, at suitably high values, it approaches the golden ratio.
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