Fibonacci Sequence Calculator
Written by Thierno Sadou Diallo, formula verified per our methodology • Last checked on 9/10/2026
The Fibonacci sequence is defined by F(0)=0, F(1)=1, and F(n)=F(n-1)+F(n-2) for all n greater than 1: each term is the sum of the two preceding ones. The 10th term of the sequence (F(10)) is 55.
Explanation
The Fibonacci sequence is one of the most famous number sequences in mathematics: it starts with 0 and 1, then each following term is obtained by adding the two terms before it (0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55...). Unlike our geometric sequence calculator, where each term depends only on the previous one multiplied by a constant ratio, the Fibonacci sequence is defined by a two-term recurrence relation: it has no simple closed formula that direct, although an explicit formula does exist (Binet's formula, based on the golden ratio). The ratio between two consecutive terms of the Fibonacci sequence gradually converges to the golden ratio (about 1.618034), a remarkable mathematical property that explains why this sequence appears in contexts as varied as the arrangement of sunflower seeds, the spiral of some mollusk shells, or the branching of some plants — though the often exaggerated ubiquity of the golden ratio in nature and art deserves to be viewed with some critical distance, many cited correspondences being approximate or anecdotal rather than rigorously proven (see our golden ratio calculator to explore that constant itself). In computing, the Fibonacci sequence is also a classic teaching example to illustrate the difference between a naive recursive calculation (potentially very slow for large values of n) and an efficient iterative one like the one used by this calculator.
Example: calculating F(10)
Inputs
Term index: 10.
Calculation
The sequence is built term by term: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55. The 10th term (counting F(0) as the first) is therefore 55.
Result
The 10th term of the Fibonacci sequence is 55.
Frequently asked questions
Why is the calculator capped at n=78?
Beyond F(78), the exact value of the Fibonacci term exceeds 2 to the power of 53 (about 9 quadrillion), the limit beyond which a standard floating-point number can no longer represent all integers exactly. Beyond that limit, the displayed result could subtly diverge from the exact mathematical value, which this calculator avoids by refusing values of n that are too high rather than showing a potentially imprecise result.
What is the link between the Fibonacci sequence and the golden ratio?
Dividing a term of the sequence by the previous term (for example F(10)÷F(9) = 55÷34 ≈ 1.6176), the result gets closer and closer to the golden ratio (about 1.618034) as n increases, without ever reaching it exactly for a finite n. This convergence is a proven mathematical property, independent of the two starting values chosen to seed a sequence of this type (0 and 1 for Fibonacci).
Why is a naive recursive calculation of Fibonacci slow for large values?
Because a naive recursive calculation (F(n) = F(n-1) + F(n-2), each call recomputing itself from scratch) recomputes the same intermediate terms a very large number of times, its complexity growing exponentially with n. An iterative calculation, like the one used by this calculator, computes each term only once, which makes it extremely fast even for high values of n.