This online Fibonacci Calculator finds the n-th term of the Fibonacci sequence using arbitrary-precision decimal arithmetic. Enter an integer number in the input field of the calculator and click the “Calculate” button.
Fibonacci sequence formula
The Fibonacci numbers, denoted \(F_n\), are the numbers that form a sequence, called the Fibonacci sequence, such that each number is the sum of the two preceding ones. The first two numbers are defined to be \(0\) and \(1\). So, for \(n>1\), we have:
$$F_0 = 0, \ F_1 = 1,$$
$$F_n = F_{n-1} + F_{n-2}.$$
So, the first 16 numbers in the sequence, from \(F_0\) to \(F_{15}\), are: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610.
The Fibonacci sequence works also for \(n \lt 0\). Using the above formulas we can get the following first 15 numbers in the sequence for \(n \lt 0\) (from \(F_{-15}\) to \(F_{-1}\)): 610, -377, 233, -144, 89, -55, 34, -21, 13, -8, 5, -3, 2, -1, 1.
For an arbitrary \(n>0\), the following general formula is valid:
$$F_{-n} = (-1)^{n+1} F_n.$$
A single Fibonacci number can be expressed using a closed-form expression known as Binet’s formula:
$$F_n = \frac{{\phi}^n \ – \ {\psi}^n}{\sqrt 5},$$
where
$$\phi = \frac{1 + {\sqrt 5}}{2} \approx 1.6180339887…$$
is the golden ratio, and \(\psi\) is its conjugate:
$$\psi = \frac{1 \ – \ {\sqrt 5}}{2} = 1 \ – \ \phi = \ – \frac{1}{\phi} \approx -0.6180339887…$$
As we can see the Fibonacci numbers are related to the golden ratio, so that the ratio of two consecutive Fibonacci numbers tends to the golden ratio as \(n\) increases.
Fibonacci numbers can often be found in various natural phenomena. Fibonacci sequences appear in biological settings, such as branching in trees, arrangement of leaves on a stem, the fruitlets of a pineapple, etc.
Fibonacci numbers appear often in mathematics. Applications of Fibonacci numbers also include computer algorithms, economics, technical analysis for financial market trading, and many more.
Our Fibonacci calculator uses arbitrary-precision decimal arithmetic, so that you can get the exact Fibonacci number even for a sufficiently large value of \(n\) within a reasonable time span (depending on the computational power of your computer).
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