Fibonacci Sequence Calculator
Find the nth Fibonacci number and the sum to n.
Results update as you type.
How it works
The Fibonacci sequence is one of the most famous patterns in mathematics, and this calculator finds any term along with related values. You enter the term position n, and it returns the nth Fibonacci number, the term immediately before it, and the sum of all terms up to n. The sequence begins 1, 1, 2, 3, 5, 8, 13, 21, and continues with each number being the sum of the previous two. It was described by Leonardo of Pisa, known as Fibonacci, in the context of a rabbit-population puzzle, but it appears far beyond that: in the arrangement of leaves and seeds in plants, the spirals of pinecones and sunflowers, and the branching of trees. The ratio of consecutive Fibonacci numbers approaches the golden ratio, about 1.618, which has fascinated mathematicians and artists for centuries. This calculator uses an efficient iterative method, so it returns results instantly even for large positions, and it also gives the running sum, useful because the sum of the first n terms equals the (n plus 2)th term minus one. Fibonacci numbers appear in computer science algorithms, financial technical analysis, and number theory. The term position must be a positive whole number, since the sequence is defined for positive indices. Enter a term position to find its Fibonacci number and the sum to that point.
F(n) = F(n−1) + F(n−2), with F(1) = F(2) = 1
Each Fibonacci number is the sum of the two before it, starting from 1 and 1. The calculator builds the sequence up to term n, returning that term, the one before it, and the running total. The sum of the first n Fibonacci numbers also equals F(n + 2) − 1, a well-known identity of the sequence.
Source:Number theory
We review all our tools from time to time to ensure accuracy.
Worked example
How to use this calculator
- 1
Enter your values
Enter term 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 series of numbers where each term is the sum of the two preceding ones, starting from 1 and 1: 1, 1, 2, 3, 5, 8, 13, and so on. It was popularized by the mathematician Fibonacci and appears widely in nature, art, and mathematics, making it one of the most recognized numerical patterns.
How is the nth Fibonacci number calculated?
You add the two previous numbers to get each new term. The calculator starts from 1 and 1 and iterates forward until it reaches position n, returning that term. This iterative approach is fast and accurate even for large positions, avoiding the rounding issues that can arise with the closed-form golden-ratio formula for very large indices.
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 1.618. This link is why Fibonacci numbers are associated with aesthetically pleasing proportions and appear in art, architecture, and design as well as in natural growth patterns.
Where does the Fibonacci sequence appear in nature?
Fibonacci numbers show up in the arrangement of leaves around stems, the spirals of sunflower seeds and pinecones, the branching of trees, and the family trees of bees, among others. These patterns arise because Fibonacci-based growth is an efficient way to pack elements or allocate resources, so evolution favors them in many organisms.
What is the sum of the first n Fibonacci numbers?
The sum of the first n Fibonacci numbers equals the (n plus 2)th Fibonacci number minus one. For example, the first 10 terms sum to 143, which is the 12th term, 144, minus one. The calculator computes this running total directly, so you can see both the individual term and the cumulative sum at once.