Fibonacci Sequence Calculator
Compute the nth Fibonacci number, generate the full sequence, and total its terms in one step.
Exact value: 55
- 1
Seed values
F(0) = 0, F(1) = 1Starting values from which every term is derived. - 2
Recurrence rule
F(k) = F(k−1) + F(k−2) for k = 2 … 10 - 3
F(10)
55
How does this calculator work?
The nth Fibonacci number follows F(n)=F(n-1)+F(n-2) starting from F(0)=0 and F(1)=1. Enter n to get that term, the full sequence from F(0) to F(n), and the running total, which equals F(n+2)-1. BigInt arithmetic keeps even very large results exact.
Formula
How this is calculated
Enter a single non-negative integer n, which is both the index of the term you want and the count that defines how long the listed sequence runs. The calculator iterates from the two seed values F(0)=0 and F(1)=1, adding the previous two terms to get each next term, so F(k)=F(k-1)+F(k-2). This bottom-up loop avoids the exponential blow-up of naive recursion and runs in linear time.
The nth Fibonacci number is simply F(n), the last term built. The sum of all listed terms F(0) through F(n) is computed using the identity Sum = F(n+2) - 1, which is exact and avoids accumulating rounding error. Because Fibonacci numbers grow roughly like the golden ratio φ≈1.618 raised to the power n, values become enormous quickly: F(100) already has 21 digits. To stay exact the engine uses arbitrary-precision BigInt arithmetic, so the displayed nth value and sum are precise even when the on-screen ResultCard number (a 64-bit float) loses precision for very large n.
Edge cases: n=0 returns F(0)=0 with the single-element sequence [0], and the sum identity still holds since F(2)-1 = 0. Inputs are floored to the nearest integer, and the range is capped at 1000 to keep the page responsive. The AreaCurve plots the numeric sequence values, visually demonstrating the exponential growth characteristic of the series.
Frequently asked questions
Yes. This calculator uses the standard zero-indexed convention where F(0)=0 and F(1)=1, so the nth term is F(n).
The hero number uses standard floating-point display, which loses precision beyond about 16 digits. The "Exact value" field uses BigInt arithmetic and is always precise.
It uses the closed identity Sum of F(0..n) = F(n+2) - 1 rather than adding every term, giving an exact result in linear time.
Also known as
TG we-Calculate Editorial Team. (2026). Fibonacci Sequence Calculator [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/fibonacci-calculator
TG we-Calculate Editorial Team. "Fibonacci Sequence Calculator." TG we-Calculate. 2026. https://we-calculate.com/calculator/fibonacci-calculator.
TG we-Calculate Editorial Team, "Fibonacci Sequence Calculator," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/fibonacci-calculator
@misc{wecalculate_fibonacci_calculator, title = {Fibonacci Sequence Calculator}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/fibonacci-calculator}}, year = {2026}, note = {TG we-Calculate} }
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