Πώς να υπολογίσετε το Fibonacci Sequence
Τι είναι το Fibonacci Sequence;
The Fibonacci sequence starts with 0 and 1, and each subsequent number is the sum of the two preceding ones. It appears throughout nature — in flower petals, spiral shells, and the golden ratio.
Τύπος
F(n) = F(n−1) + F(n−2) with F(0)=0, F(1)=1; Binet: F(n) = (φⁿ − ψⁿ)/√5 where φ=(1+√5)/2
- n
- term number in sequence
- F(n)
- the nth Fibonacci number
- φ
- golden ratio — approximately 1.618
Οδηγός βήμα προς βήμα
- 1F(0)=0, F(1)=1
- 2F(n) = F(n−1) + F(n−2)
- 3Ratio of consecutive terms approaches φ = 1.618... (golden ratio)
- 4Formula: F(n) = (φⁿ − ψⁿ)/√5 (Binet's formula)
Worked Examples
Εισαγωγή
First 10 terms
Αποτέλεσμα
0, 1, 1, 2, 3, 5, 8, 13, 21, 34
Εισαγωγή
F(20)
Αποτέλεσμα
6765
Frequently Asked Questions
Where does the Fibonacci sequence appear in nature?
Flower petals, spiral seed arrangements (sunflower), shell spirals, tree branches, and spiral galaxies all exhibit Fibonacci patterns.
What is the golden ratio and its relationship to Fibonacci?
The ratio of consecutive Fibonacci numbers approaches φ ≈ 1.618 (golden ratio). It appears in art, architecture, and nature.
Is there a closed-form formula for Fibonacci numbers?
Yes, Binet's formula: F(n) = (φⁿ − (−φ)⁻ⁿ)/√5, giving exact results for any n.
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