Golden Ratio & Fibonacci Sequence

Golden Ratio & Fibonacci Sequence

There is a special relationship between the Golden Ratio and the Fibonacci Sequence:
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, ...
(The next number is found by adding up the two numbers before it.)
And here is a surprise: when we take any two successive (one after the other) Fibonacci Numbers, their ratio is very close to the Golden Ratio.
In fact, the bigger the pair of Fibonacci Numbers, the closer the approximation. Let us try a few:
A
B
B/A
2
3
1.5
3
5
1.666666666...
5
8
1.6
8
13
1.625
...
...
...
144
233
1.618055556...
233
377
1.618025751...
...
...
...
We don't even have to start with 2 and 3, here I chose 192 and 16 (and got the sequence 192, 16, 208, 224, 432, 656, 1088, 1744, 2832, 4576, 7408, 11984, 19392, 31376, ...):
A
B
B / A
192
16
0.08333333...
16
208
13
208
224
1.07692308...
224
432
1.92857143...
...
...
...
7408
11984
1.61771058...
11984
19392
1.61815754...
...
...
...






Using The Golden Ratio to Calculate Fibonacci Numbers

And even more surprising is that we can calculate any Fibonacci Number using the Golden Ratio:
fibonacci formula phi
The answer always comes out as a whole number, exactly equal to the addition of the previous two terms.
Example:
fibonacci formula phi 6
When I used a calculator on this (only entering the Golden Ratio to 6 decimal places) I got the answer 8.00000033. A more accurate calculation would be closer to 8.





Credits: Mathisfun

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