Fibonacci Calculator — Exact F(n) with Full Digits
Find any Fibonacci number up to F(10,000) — exact digits, golden ratio approximation,
sequence generator, sum of terms & digit analysis. No rounding, no overflow.
Find the nth Fibonacci Number F(n)
F(1)=1, F(2)=1, F(n)=F(n-1)+F(n-2) · Exact BigInt arithmetic · n up to 10,000
Please enter a positive integer between 1 and 10,000
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Fibonacci Sequence
| n | F(n) first 18 digits | Digits | F(n+1)/F(n) → φ |
|---|
- What is the Fibonacci sequence?
- F(1)=1, F(2)=1, and each subsequent number is the sum of the two preceding ones: F(n) = F(n−1) + F(n−2). The sequence starts: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55…
- What is the Golden Ratio (φ)?
- φ = (1+√5)/2 ≈ 1.6180339887… As n grows, F(n+1)/F(n) converges to φ. By F(20), the ratio is accurate to 7 decimal places.
- How many digits does F(1000) have?
- F(1000) = 4346655768…8531195144 — it has exactly 209 digits. F(10000) has 2,090 digits.
- What is Cassini’s Identity?
- F(n−1)·F(n+1) − F(n)² = (−1)ⁿ. This elegant identity holds for all n ≥ 1 and can be proved by induction.
- What is the sum identity?
- The sum of the first n Fibonacci numbers equals F(n+2) − 1. For example, 1+1+2+3+5 = 12 = F(7)−1 = 13−1.
- Where does Fibonacci appear in nature?
- Spiral patterns in sunflowers, pinecones, nautilus shells, and galaxy arms closely follow Fibonacci numbers. Leaf arrangements (phyllotaxis) often have Fibonacci counts to maximize sunlight exposure.
Fibonacci numbers grow very fast, especially for higher values of n. For handling extremely large results, you can use the Big Number Calculator to work with numbers that exceed normal limits.
Fibonacci sequences are often studied alongside other math functions like permutations and factorials. If you are working with similar number patterns, try the Factorial Calculator to calculate n! values instantly.