Math
Fibonacci Calculator
The Fibonacci sequence starts 0, 1, 1, 2, 3, 5, 8, 13 and every term is the sum of the two before it.
Get any term exactly (big integers, not rounded floats) up to F(10,000), list and sum the first n terms, check whether a number belongs to the sequence, and see how the ratio of neighbours closes in on the golden ratio 1.618.
F(0) = 0, F(1) = 1, F(2) = 1, ... up to n = 10,000 (2,090 digits).
Try an example
Result
Fibonacci number
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- Position n
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- Digits
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- Sum F(0) … F(n)
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- F(n) / F(n − 1)
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Student quick launch
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Study path
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How the Fibonacci sequence works
Each term is the sum of the previous two: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, ... The numbers grow exponentially, roughly multiplying by 1.618 each step, so F(100) already has 21 digits and F(1,000) has 209.
| n | F(n) | n | F(n) |
|---|---|---|---|
| 0 | 0 | 10 | 55 |
| 1 | 1 | 12 | 144 |
| 2 | 1 | 15 | 610 |
| 3 | 2 | 20 | 6,765 |
| 5 | 5 | 30 | 832,040 |
| 8 | 21 | 50 | 12,586,269,025 |
Binet's formula
Binet's closed form gives any Fibonacci number without computing the ones before it. Because |ψ| < 1, the ψ^n term shrinks to almost nothing, so F(n) is simply φ^n / √5 rounded to the nearest whole number. Floating-point arithmetic loses exactness around n = 70, which is why this calculator uses exact big-integer doubling for the answer and shows Binet as a check.
The golden ratio connection
Divide any Fibonacci number by the one before it and the ratio approaches φ ≈ 1.6180339887. 8/5 = 1.6, 13/8 = 1.625, 21/13 = 1.615, 34/21 = 1.619. By F(20)/F(19) the ratio is correct to seven decimal places. The convergence table in the result shows this closing gap.
Useful identities
- Sum: F(0) + F(1) + ... + F(n) = F(n + 2) - 1.
- Is N Fibonacci? N is a Fibonacci number exactly when 5N² + 4 or 5N² - 4 is a perfect square.
- Every third Fibonacci number is even; every fourth is divisible by 3; every fifth by 5.
- The largest number that divides evenly into every number in a set.(F(m), F(n)) = F(gcd(m, n)).
- F(-n) = (-1)^(n+1) F(n), which extends the sequence to negative The small raised number saying how many times to multiply the base by itself..
How to use this calculator
- Pick nth number, first n numbers, or the membership check.
- Enter n (up to 10,000) or the number to test.
- Read the exact value, its digit count, the neighbours, the sum, and the ratio to the previous term.
- Open the work to see the recurrence, Binet's formula, and the perfect-square test.
Common mistakes
- Off-by-one indexing: this page uses F(0) = 0 and F(1) = 1, the standard convention. Some books start at F(1) = 1, F(2) = 1, which shifts every index by one.
- Trusting a spreadsheet past F(78): doubles lose exact integers above 2^53, so later terms are rounded.
- Assuming the ratio is exactly 1.618 for small n; it only converges to φ.
Worked examples
F(50)
The 50th Fibonacci number
Fibonacci number
12,586,269,025
F(100)
21 digits; every digit is listed in the panel
Fibonacci number
3.542248482 × 10^20
Small index
F(10) = 55
Fibonacci number
55
Zero
F(0) = 0
Fibonacci number
0
First 15 terms
0 through 377
Fibonacci number
377
Is 144 Fibonacci?
Yes, F(12)
Fibonacci number
Yes: 144 = F(12)
Is 100 Fibonacci?
No, between 89 and 144
Fibonacci number
No: 100 is not a Fibonacci number
Large check
12586269025 = F(50)
Fibonacci number
Yes: 12,586,269,025 = F(50)
Out of range
n above 10,000
Fibonacci number
—
Frequently asked questions
What is the Fibonacci sequence?+
0, 1, 1, 2, 3, 5, 8, 13, 21, ... where each number is the sum of the two before it. It is defined by F(0) = 0, F(1) = 1, F(n) = F(n - 1) + F(n - 2).
What is the 100th Fibonacci number?+
F(100) = 354,224,848,179,261,915,075, a 21-digit number. This page computes it exactly rather than as a rounded decimal.
How do I check if a number is a Fibonacci number?+
Compute 5N² + 4 and 5N² - 4. If either is a perfect square, N is Fibonacci. For N = 144: 5 × 144² - 4 = 103,676 = 322², so yes.
What is Binet's formula?+
F(n) = (φ^n - ψ^n) / √5 with φ = 1.618... and ψ = -0.618.... It gives Fibonacci numbers directly, but needs exact arithmetic for large n.
Why do Fibonacci ratios approach the golden ratio?+
Dividing the recurrence by F(n - 1) gives r = 1 + 1/r in the limit, whose positive solution is φ = (1 + √5)/2.
Is 0 a Fibonacci number?+
Yes, F(0) = 0 in the standard indexing. Some older texts start at F(1) = 1 and skip it.
How fast do Fibonacci numbers grow?+
Exponentially, by a factor of about 1.618 per step. The number of digits grows by roughly 0.209 per term, so F(10,000) has 2,090 digits.
About this calculator
- Written by
- mathcheck editorial team
- Last reviewed
- September 4, 2026
Method
- Uses the values entered by the user and stable formulas documented on the page.
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Last updated: September 4, 2026