About the Fibonacci Calculator
This Fibonacci calculator returns the exact nth term of the Fibonacci sequence — 0, 1, 1, 2, 3, 5, 8, 13, … — where each number is the sum of the two before it. It uses whole-number (big integer) arithmetic, so even F(1000), a 209-digit number, is shown digit for digit rather than rounded.
Along with the nth term you get the full sequence up to n (the first 100 terms are listed), the sum of all terms from F(0) to F(n), the number of digits, and the ratio of consecutive terms, which settles on the golden ratio φ ≈ 1.618034. It is handy for maths homework, programming exercises, puzzle checking and anyone curious about Fibonacci numbers in nature and design.
The calculator follows the standard convention F(0) = 0 and F(1) = 1. You can also generate the Lucas numbers (2, 1, 3, 4, 7, …) or any Fibonacci-like sequence by choosing your own two starting values.
How to use the fibonacci calculator
- 1Enter the term number n you want (0 to 5,000).
- 2Keep the standard Fibonacci sequence, or pick Lucas or custom starting values.
- 3Read the exact nth term and copy it with the copy button.
- 4Scroll the sequence list and the ratio chart to see the pattern.
Formula and method
Each Fibonacci number is the sum of the two before it. The calculator builds the sequence term by term with exact integer arithmetic, so there is no rounding error however large n is. Lucas numbers use the same rule with starting values 2 and 1, and custom sequences use any two whole numbers you choose.
Binet’s formula F(n) = (φⁿ − ψⁿ) / √5, with φ = (1 + √5)/2 and ψ = (1 − √5)/2, gives the same values and explains why the ratio F(n)/F(n−1) converges to the golden ratio φ ≈ 1.6180339887. The running total obeys the identity F(0) + F(1) + … + F(n) = F(n+2) − 1.
- F(n)
- The nth Fibonacci number (starting from F(0) = 0)
- φ
- Golden ratio, (1 + √5) ÷ 2 ≈ 1.6180339887
- n
- Position (index) in the sequence
Worked examples
The 10th Fibonacci number
The sequence runs 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, so F(10) = 55. The terms F(0) through F(10) add up to 143, which is F(12) − 1 = 144 − 1, and 55 ÷ 34 ≈ 1.6176 is already close to φ.
F(100) exactly
The 100th Fibonacci number is 354,224,848,179,261,915,075, a 21-digit integer. Standard floating-point arithmetic can only hold about 15–16 significant digits, which is why the calculator uses exact big-integer addition.
The 10th Lucas number
Lucas numbers start 2, 1 and follow the same rule: 2, 1, 3, 4, 7, 11, 18, 29, 47, 76, 123. So L(10) = 123, the first eleven terms sum to 321, and 123 ÷ 76 ≈ 1.6184.
Frequently asked questions
What is the Fibonacci sequence?+
It is the series 0, 1, 1, 2, 3, 5, 8, 13, 21, … in which every number is the sum of the previous two. It is named after Leonardo of Pisa (Fibonacci), who described it in 1202 in a problem about breeding rabbits.
Does the Fibonacci sequence start with 0 or 1?+
Modern mathematics usually starts at F(0) = 0 and F(1) = 1, which this calculator uses. Some textbooks start at F(1) = F(2) = 1; the numbers are the same, only the index shifts by one.
How is the Fibonacci sequence related to the golden ratio?+
The ratio of consecutive Fibonacci numbers, such as 89/55 or 144/89, gets closer and closer to the golden ratio φ ≈ 1.6180339887. Binet’s formula shows F(n) is the nearest integer to φⁿ ÷ √5.
What is the 100th Fibonacci number?+
F(100) = 354,224,848,179,261,915,075 when counting from F(0) = 0. It has 21 digits. Enter any n up to 5,000 above to get the exact value.
What are Lucas numbers?+
Lucas numbers follow the same addition rule as Fibonacci numbers but start with 2 and 1: 2, 1, 3, 4, 7, 11, 18, … They also approach the golden ratio and satisfy L(n) = F(n−1) + F(n+1).
What is the sum of the first n Fibonacci numbers?+
The sum F(0) + F(1) + … + F(n) equals F(n+2) − 1. For example, 0+1+1+2+3+5+8 = 20 = F(8) − 1 = 21 − 1.