কীভাবে Fibonacci Sequence গণনা করবেন
Fibonacci Sequence কি?
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.
সূত্র
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
ধাপে ধাপে নির্দেশিকা
- 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)
সমাধান করা উদাহরণ
ইনপুট
First 10 terms
ফলাফল
0, 1, 1, 2, 3, 5, 8, 13, 21, 34
ইনপুট
F(20)
ফলাফল
6765
সচরাচর জিজ্ঞাসা
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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