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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.
ସୂତ୍ର
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
ଷ୍ଟେପ୍-ଷ୍ଟେପ୍ ଗାଇଡ୍ |
- 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)
ସମାଧାନ ହୋଇଥିବା ଉଦାହରଣ
ଇନପୁଟ୍
First 10 terms
ଫଳ
0, 1, 1, 2, 3, 5, 8, 13, 21, 34
ଇନପୁଟ୍
F(20)
ଫଳ
6765
ବାରମ୍ବାର ଜିଜ୍ଞାସା
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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