Fibonacci Calculator
10
F(10)34
Sequence
0, 1, 1, 2, 3, 5, 8, 13, 21, 34
The Fibonacci sequence is a famous series of numbers where each term is the sum of the two before it, starting from 0 and 1: 0, 1, 1, 2, 3, 5, 8, 13, and so on. This calculator returns the value at any position n and lists the opening terms of the sequence. Fibonacci numbers appear throughout mathematics, nature, and computer science, and their ratios converge toward the golden ratio.
Formula
F(n) = F(n-1) + F(n-2), with F(1) = 0 and F(2) = 1
- F(n)
- The Fibonacci number at position n
- n
- The position in the sequence (positive integer, 1 to 100)
- F(n-1), F(n-2)
- The two preceding Fibonacci numbers
How it works
- Enter the position n (a positive integer up to 100). The calculator treats position 1 as the value 0, position 2 as 1, and so on.
- The engine builds the sequence iteratively by repeatedly adding the previous two terms, then returns the value at position n along with the first terms of the series for context.
- Read the nth Fibonacci number in the result, and review the listed sequence (the first 20 terms are shown) to see the pattern leading up to it.
Worked example
Find the 10th Fibonacci number.
- Start the sequence: 0, 1, 1, 2, 3 (positions 1 through 5).
- Continue adding the last two terms: 5, 8, 13, 21 (positions 6 through 9).
- The 10th term is 21 + 13 = 34.
The 10th Fibonacci number is 34, with the sequence 0, 1, 1, 2, 3, 5, 8, 13, 21, 34.
Frequently asked questions
- Does this calculator start the sequence at 0 or 1?
- It starts at 0. Position 1 returns 0, position 2 returns 1, position 3 returns 1, and so on. This is the standard zero-indexed convention used in most mathematical references.
- What is the largest position I can enter?
- You can request up to position 100. Beyond that, the values exceed the precision limits of standard floating-point numbers, so the calculator caps the input to keep results reliable.
- How is the Fibonacci sequence related to the golden ratio?
- As you go further along the sequence, the ratio of each term to the one before it gets closer and closer to the golden ratio, approximately 1.618. This relationship is why Fibonacci numbers appear in geometry and design.
- Where do Fibonacci numbers appear in real life?
- They show up in the spiral arrangement of sunflower seeds, pinecones, and shells, in branching patterns of plants, and in computer science algorithms such as the Fibonacci search technique and certain data structures.