FibLab

Fibonacci Calculator

F(n) = F(n−1) + F(n−2).

Index n (0-based)

F(n)

F(n) = F(n−1) + F(n−2)

F(n+1): 89
Golden ratio approach: 1.618

What you entered

Index n (0-based): 10

1
Build the sequence from F(0)=0, F(1)=1
F(k) = F(k−1) + F(k−2)= 0, 1, 1, 2, 3, 5, 8, 13, …
2
F(10)
F(10)= 55
3
F(11)
F(11)= 89
4
Ratio approaches the golden ratio φ
F(11) ÷ F(10)= 1.6182

Result

F(n): 55

The 10th Fibonacci number is 55. As n grows, the ratio of consecutive terms converges toward the golden ratio, ≈1.618.

Fibonacci numbers and the golden ratio

Each term is the sum of the two before it. The ratio column divides the following term by the current one, which converges on the golden ratio.

n	F(n)	Next term	Ratio of consecutive terms
5	5	8	1.600000
8	21	34	1.619048
10	55	89	1.618182
12	144	233	1.618056
15	610	987	1.618033
20	6,765	10,946	1.618034
25	75,025	121,393	1.618034
30	832,040	1,346,269	1.618034

The last column is the point of the table: the ratio oscillates above and below the golden ratio and converges on 1.618034 by about the fifteenth term, and it does so for any starting pair, not just 1 and 1. The sequence appears in the spiral arrangement of sunflower seeds and pine cones because packing seeds at the golden angle leaves the fewest gaps - it is an efficiency result, not mysticism. Growth is roughly exponential, multiplying by about 1.618 each step, so F(30) is already over 800,000.

Each number is the sum of the two before it

Starting from 0 and 1, every Fibonacci number is generated by adding the previous two: 0, 1, 1, 2, 3, 5, 8, 13, 21… — one of the simplest rules in math that produces surprisingly rich patterns.

Where it shows up in nature

Sunflower seed spirals, pinecone scales, and nautilus shells all approximate Fibonacci-based growth patterns, and the ratio between consecutive terms converges toward the golden ratio (≈1.618) as n grows.

Frequently asked questions

What are Fibonacci numbers?

The sequence 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144… where each number is the sum of the two before it. F(0) = 0, F(1) = 1, and F(n) = F(n−1) + F(n−2). The sequence appears throughout mathematics and nature.

What is the connection between Fibonacci and the golden ratio?

The ratio of consecutive Fibonacci numbers (F(n+1)/F(n)) converges to the golden ratio φ ≈ 1.6180339887. By F(10), the ratio is already 89/55 = 1.6181… accurate to 3 decimal places. φ = (1 + √5) / 2.

Where do Fibonacci numbers appear in nature?

Flower petals (lilies have 3, buttercups 5, daisies 34 or 55), sunflower seed spirals (usually 34 and 55 spirals), pinecone scales, pineapple hexagons, and tree branching patterns. The connection is likely because Fibonacci packing is the most efficient growth strategy.