Fibonacci (n-ésimo termo)
- Created by
- Renato Passos, Eng. de Software
- Reviewed by
- Renato Passos, Eng. de Software
Last updated: Apr 18, 2026
Formula
recorrência
About this calculator
The Fibonacci n-th term calculator determines the value of F_n, where F_0 = 0, F_1 = 1, and each subsequent term is the sum of the previous two. The sequence starts 0, 1, 1, 2, 3, 5, 8, 13... and is widely found in nature, art, and computing.
The calculation is based on the recursive definition: F(n) = F(n-1) + F(n-2). For large n, the program uses dynamic programming or Binet's formula to avoid excessive recursion. The result is exact for integers within numerical precision limits.
Use this calculator to quickly find a specific term of the sequence. For example, to find the 10th term (F_10 = 55) or the 20th (F_20 = 6765). It is useful in algorithm analysis, modeling rabbit population growth, or simple mathematical curiosity.
Caution: for very large n (above 78), the number may exceed 64-bit integer capacity, resulting in approximations. Also, the sequence grows exponentially, so high terms are huge. Check if the result seems reasonable.
Frequently asked questions
What is the formula for the n-th Fibonacci term?
The recursive formula is F(n) = F(n-1) + F(n-2), with F(0)=0 and F(1)=1. There is also Binet's formula using the golden ratio.
How to calculate Fibonacci for negative n?
The Fibonacci sequence is defined only for non-negative n. For negative n, use the negative extension: F(-n) = (-1)^(n+1) * F(n).
What is the largest term I can calculate accurately?
On 64-bit systems, the largest exact term is F(78) = 8944394323791464. Beyond that, overflow or loss of precision occurs.
What is the Fibonacci sequence used for in daily life?
It is used in search algorithms, financial market analysis, design proportions, and in nature (sunflower spirals, shells).
Can the result be negative?
No, for n >= 0 all terms are non-negative. The sequence starts with 0 and increases.