Generate the Fibonacci sequence up to any term, find a specific term's value, and see how it approaches the golden ratio.
The Fibonacci sequence starts with 0 and 1, and every following term is the sum of the two terms before it: 0, 1, 1, 2, 3, 5, 8, 13, 21... As the sequence grows, the ratio between consecutive terms gets closer and closer to the golden ratio, approximately 1.6180339887.
F(0) = 0, F(1) = 1
F(n) = F(n−1) + F(n−2) for n ≥ 2
Enter which term number you want the value of (n starts at 0), and how many terms of the sequence you'd like listed. The calculator generates the sequence and shows the golden ratio approximation using the last two generated terms.
For n = 10, the 10th term (0-indexed) is 55, following the sequence 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55. The ratio 55 ÷ 34 ≈ 1.6176, already very close to the golden ratio of about 1.618.
It's a series of numbers where each number is the sum of the two preceding ones, starting 0, 1, 1, 2, 3, 5, 8, 13, and so on.
Either by iterating the sequence up to the desired position, or using Binet's closed-form formula involving the golden ratio for a direct calculation.
It's often observed in spiral patterns like sunflower seed heads, pinecones, and nautilus shells, though the connection is sometimes overstated in popular culture.