How the golden ratio and the Fibonacci sequence are connected

Recall that the golden ratio is a special irrational number that is approximately equal to 1.618.  It appears frequently in geometry, art, and architecture.

Also recall that the Fibonacci sequence is the series of numbers:

0, 1, 1, 2, 3, 5, 8, 13, 21, 34 …

Each number is found by adding up the two numbers before it.  For example 5 is found by adding 3 and 2.

German Astronomer Johannes Kepler once wrote that “as 5 is to 8, so 8 is to 13, approximately, and as 8 is to 13, so 13 is to 21, approximately”.  What Kepler is saying is that the ratios of consecutive numbers in the Fibonacci sequence are similar.

About a century after Kepler’s statement Scottish mathematician Robert Simson discovered that if you take the ratios of consecutive numbers in the Fibonacci sequence, and put them in the sequence

1/1, 2/1, 3/2, 5/3, 8/5, 13/8, 21/13, 34/21, 55/34 …

or to three decimal places

1, 2, 1.5, 1.667, 1.6, 1.625, 1.615, 1.619, 1.618 ….

the numbers of these terms get closer and closer to the golden ratio.

What Simson discovered is that the golden ratio is approximated by the ratio of consecutive numbers in the Fibonacci sequence, with the accuracy of the approximation increasing as we move down the sequence.

Now here’s something else, lets consider a Fibonacci like sequence starting with two random numbers and them adding consecutive terms to continue the sequence.  So, lets start with 4 and 10, the next term would be 14 and the next 24.  Our example would give us:

4, 10, 14, 24, 38, 62, 100, 162, 262, 424 …

lets check the ratios of consecutive terms …

10/4, 14/10, 24/14, 38/24, 62/38, 100/62, 162/100, 262/162, 424/262 …

or to three decimal places

2.5, 1.4, 1.714, 1.583, 1.632, 1.612, 1.620, 1.617, 1.618 …

Try this with any other two terms and you will see the Fibonacci recurrence algorithm of adding two consecutive terms in a sequence to get the next term is so powerful that whatever two numbers you start with, the ratio of consecutive terms will always converge to the golden ratio!

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