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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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Q: What does Fibonacci and the golden ratio have in common?

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No, but the ratio of each term in the Fibonacci sequence to its predecessor converges to the Golden Ratio.

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.

with a Fibonacci gauge

The golden ratio is approximately 1.618: 1. This ratio is commonly found in nature and architecture. Stock traders often look for this ratio in patterns on stock charts. One way to compute this ratio is to compare any adjacent Fibonacci numbers. For this reason stock traders often refer to this type of analysis using the term Fibonacci, as in "Fibonacci retracements".

The pentagram is related to the golden ratio, because the diagonals of a pentagram sections each other in the golden ratio. The Fibonacci numbers are also related to the golden ratio. Take two following Fibonacci numbers and divide them. So you have 2:1, 3:2, 5:3, 8:5 and so on. This sequence is going to 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.

The limit of the ratio is the Golden ratio, or [1 + sqrt(5)]/2

Your mind will be blown if you search Phi, The golden ratio, or the fibonacci sequence. It has to do with everything.

Fibonacci didn't discover the golden ratio. It had been used thousands of years earlier,for example in construction of religious architecture by the Greeks, who considered it themost perfect and visually pleasing ratio of structural length to width. Fibonacci studied asimple numerical series that generates the number equal to the golden ratio.The number is also the solution to the equation: [ (x - 1) = 1/x ].

The 'golden ratio' is the limit of the ratio of two consecutive terms of the Fibonacci series, as the series becomes very long. Actually, the series converges very quickly ... after only 10 terms, the ratio of consecutive terms is already within 0.03% of the golden ratio.

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