Fibonacci Calculator
Find the nth Fibonacci number and the running sum.
F₁ = 1, F₂ = 1, up to n = 90
Results update as you type.
How it works
The Fibonacci sequence is one of the most famous patterns in mathematics: it begins with 1 and 1, and each following number is the sum of the two before it, giving 1, 1, 2, 3, 5, 8, 13, 21, and so on. This Fibonacci calculator returns the number at any position n up to the ninetieth term, together with the sum of all the terms up to that point. Despite its simple rule, the sequence appears throughout nature and design, from the spiral arrangement of sunflower seeds and pinecone scales to the branching of trees, and the ratio of consecutive Fibonacci numbers steadily approaches the golden ratio of roughly 1.618. Programmers meet it as the classic example for teaching recursion and dynamic programming, traders reference Fibonacci retracement levels, and students study it in number theory and combinatorics. The calculator uses an efficient iterative method that walks the sequence forward once, so even the ninetieth term is returned instantly without the exponential slowdown that a naive recursive approach would suffer. Enter a position and you immediately see both that Fibonacci number and the accumulated total.
F(n) = F(n−1) + F(n−2), with F(1) = F(2) = 1
The Fibonacci sequence starts with two ones, and every term after that is the sum of the two terms immediately before it, so the series grows one and one to two, three, five, eight, thirteen, and onward. The calculator builds the sequence iteratively up to the position you request, avoiding the slow repeated recomputation of a naive recursive definition, and it accumulates a running total so the sum of the first n terms is available at the same time.
Source:Fibonacci recurrence relation
We review all our tools from time to time to ensure accuracy.
Worked example
How to use this calculator
- 1
Enter your values
Enter position (n).
- 2
Adjust the options
Fine-tune any options to match your situation.
- 3
Read your result
Nth Fibonacci number updates instantly as you type — no button to press.
Frequently asked questions
What is the Fibonacci sequence?
The Fibonacci sequence is a list of numbers where each term is the sum of the two preceding ones, starting from 1 and 1. That gives 1, 1, 2, 3, 5, 8, 13, 21, 34, and so on without end. It was popularised in Europe by Leonardo of Pisa, known as Fibonacci, and it models many growth patterns found in nature.
How is the nth Fibonacci number found?
You can add the previous two numbers repeatedly until you reach the position you want, which is exactly what the calculator does internally in a single efficient pass. There is also a closed-form expression called Binet's formula using the golden ratio, but for exact integer results the iterative addition method is simpler and avoids rounding errors at large positions.
What is the connection to the golden ratio?
As you go further along the Fibonacci sequence, the ratio of each number to the one before it gets closer and closer to the golden ratio, approximately 1.61803. Divide 13 by 8 and you get 1.625; divide 89 by 55 and you get about 1.618. This convergence links the discrete sequence to the golden ratio that appears widely in art, architecture, and geometry.
Where does Fibonacci appear in nature?
Fibonacci numbers show up in the number of petals on many flowers, the spiral counts of sunflower seeds and pinecone scales, the branching of trees, and the arrangement of leaves around a stem. These patterns arise because packing elements at the golden angle, which comes from the same ratio, is an efficient way to fill space and capture light without overlap.
Why does this calculator stop at n = 90?
Around the ninety-third Fibonacci number the value exceeds the largest integer that standard computer arithmetic can represent exactly, so results beyond that point would lose precision and become unreliable. Capping the input at ninety guarantees every returned term and sum is an exact integer. That range already covers essentially every practical and educational use of the sequence.