Fibonacci Sequence - Nature's Own Number Pattern
The Fibonacci sequence is one of the most famous number patterns in all of mathematics. It appears in flower petals, spiral shells, pine cones, and the proportions of living things. Despite its simple rule, it connects to an astonishing range of mathematical ideas.
The Rule
Start with 1 and 1. Each new term is the sum of the two terms before it.
F1 = 1, F2 = 1, Fn = Fn−1 + Fn−2 for n ≥ 3.
The First Twenty Terms
| n | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
|---|---|---|---|---|---|---|---|---|---|---|
| Fn | 1 | 1 | 2 | 3 | 5 | 8 | 13 | 21 | 34 | 55 |
| n | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 |
|---|---|---|---|---|---|---|---|---|---|---|
| Fn | 89 | 144 | 233 | 377 | 610 | 987 | 1,597 | 2,584 | 4,181 | 6,765 |
The Golden Ratio
As you go further along the Fibonacci sequence, the ratio of consecutive terms gets closer and closer to a special number called the Golden Ratio, denoted by the Greek letter phi (φ).
φ = (1 + √5) / 2 ≈ 1.6180339…
For example: 89/55 ≈ 1.6182, 144/89 ≈ 1.6180. The ratio converges to φ rapidly.
Fibonacci in Nature
| Where | How Fibonacci appears |
|---|---|
| Sunflower seeds | Spirals in opposite directions: typically 34 and 55 (consecutive Fibonacci numbers) |
| Flower petals | Most flowers have 3, 5, 8, 13, or 21 petals |
| Pine cones | Spiral rows: usually 8 and 13 |
| Nautilus shell | Each chamber approximately 1.618 times the size of the previous one |
| Human body | Proportions of fingers, forearm to hand, approximate the Golden Ratio |
Interesting Properties
- Every third Fibonacci number is even.
- Every fourth Fibonacci number is a multiple of 3.
- The sum of the first n Fibonacci numbers equals Fn+2 − 1.
- The square of any Fibonacci number equals the product of its two neighbours, plus or minus 1 (Cassini's identity).
Worked Examples
F11 = 34 + 55 = 89. F12 = 55 + 89 = 144.
Sum = F12 − 1 = 144 − 1 = 143.
Verify: 1+1+2+3+5+8+13+21+34+55 = 143 ✓
Key Takeaways
- Fibonacci rule: each term = sum of the two before it. Fn = Fn−1 + Fn−2.
- Sequence: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, …
- The ratio of consecutive terms converges to the Golden Ratio φ ≈ 1.618.
- Fibonacci numbers appear throughout nature in spirals, petals, and proportions.
Practice Questions
- Write the next four terms after: 1, 1, 2, 3, 5, 8, 13, 21, …
- The 12th Fibonacci number is 144 and the 13th is 233. Find the 14th and 15th.
- Calculate 89 ÷ 55 and 144 ÷ 89. How close are these to the Golden Ratio?
- Use the sum shortcut to find the sum of the first 12 Fibonacci numbers.
- A sequence follows the Fibonacci rule but starts with 3 and 4. Write the first eight terms.