What is the Fibonacci Series?
The Fibonacci series (or Fibonacci sequence) is a sequence of numbers where each number is the sum of the two preceding numbers. Starting with 0 and 1, the sequence goes: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144... and continues infinitely.
The Fibonacci Formula
F(n) = F(n-1) + F(n-2)
where F(0) = 0 and F(1) = 1
This recursive formula says: to find any Fibonacci number, add the previous two Fibonacci numbers together.
Step-by-Step: Finding F(10)
F(0) = 0 (starting value)
F(1) = 1 (starting value)
F(2) = F(1) + F(0) = 1 + 0 = 1
F(3) = F(2) + F(1) = 1 + 1 = 2
F(4) = F(3) + F(2) = 2 + 1 = 3
F(5) = F(4) + F(3) = 3 + 2 = 5
F(6) = F(5) + F(4) = 5 + 3 = 8
F(7) = F(6) + F(5) = 8 + 5 = 13
F(8) = F(7) + F(6) = 13 + 8 = 21
F(9) = F(8) + F(7) = 21 + 13 = 34
F(10) = F(9) + F(8) = 34 + 21 = 55
Therefore, F(10) = 55
The Golden Ratio Connection
When you divide consecutive Fibonacci numbers, the ratio approaches the Golden Ratio (φ ≈ 1.618034...). This beautiful number appears throughout mathematics, art, and nature.
8 ÷ 5 =
1.600
13 ÷ 8 =
1.625
21 ÷ 13 =
1.615
89 ÷ 55 =
1.618
As numbers get larger, the ratio converges to φ = (1 + √5) / 2 ≈ 1.6180339887...
Binet's Formula (Direct Calculation)
Instead of calculating all previous terms, you can find the nth Fibonacci number directly using Binet's formula:
F(n) = (φⁿ - ψⁿ) / √5
where φ = (1 + √5) / 2 ≈ 1.618 and ψ = (1 - √5) / 2 ≈ -0.618
For large n, since |ψ| < 1, ψⁿ becomes negligible, so F(n) ≈ φⁿ / √5 (rounded to nearest integer).
Fibonacci in Nature
The Fibonacci sequence appears remarkably often in nature, a phenomenon that has fascinated mathematicians and scientists for centuries:
Flower Petals
- • Lilies have 3 petals
- • Buttercups have 5 petals
- • Delphiniums have 8 petals
- • Marigolds have 13 petals
- • Daisies often have 34, 55, or 89 petals
Spirals in Nature
- • Sunflower seed heads: 34 and 55 spirals
- • Pinecone scales form Fibonacci spirals
- • Pineapple scales: 8, 13, and 21 spirals
- • Nautilus shell curves approach golden spiral
- • Hurricane and galaxy spirals
Plant Growth
- • Branch arrangement on trees
- • Leaf arrangement (phyllotaxis)
- • Root system patterns
- • Artichoke florets
Animal Kingdom
- • Honeybee family tree
- • Starfish arms (5)
- • Sand dollars (5)
- • Reproductive patterns
Real-World Usage
Fibonacci Trading (Stock Market)
In Fibonacci trading, traders use Fibonacci retracement levels (23.6%, 38.2%, 50%, 61.8%) to predict support and resistance. If a stock rose from $100 to $200, what are the Fibonacci retracement levels?
The retracement levels come from Fibonacci ratios:
• 23.6% = 1 - (F(n)/F(n+3)) as n → ∞
• 38.2% = 1 - 0.618 = 1/φ²
• 61.8% = 1/φ (the golden ratio inverse)
For a $100 rise ($100 → $200):
• 23.6% retracement: $200 - ($100 × 0.236) = $176.40
• 38.2% retracement: $200 - ($100 × 0.382) = $161.80
• 50.0% retracement: $200 - ($100 × 0.500) = $150.00
• 61.8% retracement: $200 - ($100 × 0.618) = $138.20
Important Properties of Fibonacci Numbers
1. Sum Property
F(0) + F(1) + F(2) + ... + F(n) = F(n+2) - 1
2. Cassini's Identity
F(n-1) × F(n+1) - F(n)² = (-1)ⁿ
3. GCD Property
GCD(F(m), F(n)) = F(GCD(m, n))
4. Divisibility Patterns
Every 3rd Fibonacci is divisible by 2, every 4th by 3, every 5th by 5, every 6th by 8...
5. Zeckendorf's Theorem
Every positive integer can be uniquely represented as a sum of non-consecutive Fibonacci numbers.
First 30 Fibonacci Numbers
| n | F(n) |
|---|---|
| 0 | 0 |
| 1 | 1 |
| 2 | 1 |
| 3 | 2 |
| 4 | 3 |
| 5 | 5 |
| 6 | 8 |
| 7 | 13 |
| 8 | 21 |
| 9 | 34 |
| 10 | 55 |
| 11 | 89 |
| 12 | 144 |
| 13 | 233 |
| 14 | 377 |
| 15 | 610 |
| 16 | 987 |
| 17 | 1597 |
| 18 | 2584 |
| 19 | 4181 |
| 20 | 6,765 |
| 21 | 10,946 |
| 22 | 17,711 |
| 23 | 28,657 |
| 24 | 46,368 |
| 25 | 75,025 |
| 26 | 121,393 |
| 27 | 196,418 |
| 28 | 317,811 |
| 29 | 514,229 |