Fibonacci Sequence Generator
Generate the Fibonacci sequence up to any term n. Shows the ratio between consecutive terms approaching the golden ratio (φ). Useful for programming and math education.
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Applications
Used by programming students verifying iterative and recursive implementations, by math teachers demonstrating the convergence to the golden ratio, and by number-theory enthusiasts exploring Lucas, Tribonacci, and other generalized recurrences.
How It Works
The standard Fibonacci sequence starts with F(0)=0 and F(1)=1, where each term is the sum of the two preceding ones: F(n)=F(n-1)+F(n-2). This tool uses BigInt-based iterative computation to support large n, and reports the golden ratio approximation F(n+1)/F(n) which converges to phi = (1 + sqrt(5))/2 ≈ 1.6180339887 as n grows. Binet's closed form F(n) = (phi^n - psi^n)/sqrt(5) (where psi = (1 - sqrt(5))/2) is shown for reference.
Fun Fact: Leonardo of Pisa (Fibonacci) introduced the sequence to Western European mathematics in 1202 in his book Liber Abaci, but Indian mathematicians like Pingala had described the same recurrence centuries earlier — and the ratio F(n+1)/F(n) converging to phi was known to the Greeks through the pentagram.
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