Here's why and how:
The Connection:
* The Fibonacci sequence is defined by the pattern: 0, 1, 1, 2, 3, 5, 8, 13, 21... where each number is the sum of the two preceding ones.
* As you move further down the sequence, the ratio of consecutive Fibonacci numbers gets closer and closer to the golden ratio.
Calculation:
1. Choose any two consecutive Fibonacci numbers. For example, 8 and 13.
2. Divide the larger number by the smaller number. 13 / 8 = 1.625
3. Repeat this process with the next two consecutive Fibonacci numbers. For example, 13 and 21. 21 / 13 = 1.615
4. Continue this process with larger and larger Fibonacci numbers. You'll notice that the ratio gets closer and closer to approximately 1.6180339887.
The Limit:
* Mathematically, this means the golden ratio is the limit of the ratio of consecutive Fibonacci numbers as n approaches infinity:
* φ = lim (n→∞) F(n+1) / F(n)
Important Note:
While the Fibonacci sequence provides a way to *approximate* the golden ratio, it doesn't provide an exact value. To get the precise value of φ, you would use the mathematical formula:
* φ = (1 + √5) / 2