Fibonacci Sequence Calculator

Generate Fibonacci sequence up to 100 terms, analyze convergence to the golden ratio, and directly calculate specific nth terms using Binet's formula.
Fibonacci Sequence
Calculation Result
Binet's Formula
Binet's formula is a closed-form expression that directly calculates the nth Fibonacci number without recursion:

F(n) = (φⁿ - ψⁿ) / √5

Where:
• φ (phi) = (1 + √5) / 2 ≈ 1.618033988749 (golden ratio)
• ψ (psi) = (1 - √5) / 2 ≈ -0.618033988749
• √5 ≈ 2.236067977500

This formula allows O(1) time complexity calculation of the nth term.
Golden Ratio Convergence (Last 20 Terms)
n F(n) F(n+1) Fn+1/Fn Difference from φ
Fibonacci in Nature
The Fibonacci sequence appears remarkably often in nature:

1. Plant Leaf Arrangements
• Sunflower seeds: Spirals in clockwise and counterclockwise directions are Fibonacci numbers (34, 55, 89, etc.)
• Pine cones: Spiral patterns of 8 and 13, or 5 and 8
• Pineapples: Scales in spirals of 8, 13, 21

2. Petal Counts
• Lily: 3 petals
• Buttercup: 5 petals
• Cosmos: 8 petals
• Marigold: 13 petals
• Daisy: 21, 34 petals

3. Shells
• Nautilus shell spirals follow the golden ratio (1.618...)

4. Human Body
• Finger bone length ratios
• Ratio of entire arm to forearm
• Facial proportions (chin to nose, nose to eyebrows)

Complete Fibonacci Sequence Guide: Mathematics, Nature, and Golden Ratio (2025)

Definition and Basic Concepts of Fibonacci Sequence

The Fibonacci Sequence is a series of numbers introduced by Italian mathematician Leonardo Fibonacci in 1202 in his book "Liber Abaci", where each term is defined as the sum of the two preceding terms. The sequence starts with F(0) = 0, F(1) = 1, and follows the recurrence relation F(n) = F(n-1) + F(n-2). The first 10 terms are 0, 1, 1, 2, 3, 5, 8, 13, 21, 34. For example, F(5) = F(4) + F(3) = 3 + 2 = 5, F(6) = F(5) + F(4) = 5 + 3 = 8. While originating from a rabbit breeding problem, this sequence appears in mathematics, nature, art, and computer science. Fibonacci numbers grow exponentially, approximating φⁿ / √5 for large n (φ is the golden ratio 1.618...).

Golden Ratio (φ) and Its Relationship with Fibonacci

The ratio of consecutive Fibonacci numbers F(n+1)/F(n) converges to the golden ratio φ = (1 + √5) / 2 ≈ 1.618033988749.... For example, F(6)/F(5) = 8/5 = 1.6, F(10)/F(9) = 55/34 = 1.6176..., F(20)/F(19) = 6765/4181 = 1.6180339..., getting closer to φ. As n approaches infinity, this ratio converges exactly to φ. The golden ratio, known since ancient Greece as the most aesthetically pleasing proportion, appears in the Parthenon, pyramids, and Leonardo da Vinci's "Mona Lisa" and "Last Supper".

Calculating nth Term with Binet's Formula

Binet's Formula, discovered by French mathematician Jacques Philippe Marie Binet, is a closed-form expression that directly calculates the nth Fibonacci number without recursion: F(n) = (φⁿ - ψⁿ) / √5. Where φ = (1 + √5) / 2 ≈ 1.618033988749 (golden ratio), ψ = (1 - √5) / 2 ≈ -0.618033988749. While recursive methods require billions of operations for F(50) (O(2ⁿ) time), Binet's formula computes it in one calculation (O(1) time).

Fibonacci in Nature: Patterns in Plants and Life

The Fibonacci sequence appears remarkably throughout nature. Most notably in plant leaf arrangements (phyllotaxis), sunflower seeds are arranged in spirals of 34 clockwise and 55 counterclockwise (or 55 and 89), all Fibonacci numbers. This pattern maximizes space efficiency and sunlight distribution. Pine cone scales form 8 and 13 spirals, pineapples have 8, 13, 21 spirals. Petal counts also follow Fibonacci: lilies 3, buttercups 5, cosmos 8, marigolds 13, daisies 21, 34.

Mathematical Properties and Identities of Fibonacci

The Fibonacci sequence has many fascinating mathematical properties. Cassini's Identity: F(n-1) × F(n+1) - F(n)² = (-1)ⁿ. For n=5: 3 × 8 - 5² = 24 - 25 = -1 = (-1)⁵. Sum formula: F(0) + F(1) + ... + F(n) = F(n+2) - 1. Sum of first 10 terms: 0+1+1+2+3+5+8+13+21+34 = 88 = F(12) - 1. GCD property: gcd(F(m), F(n)) = F(gcd(m, n)).