Calculate the Fibonacci term or nth value from Binet’s formula by entering n or F(n) to find the missing Fibonacci result quickly and easily.
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Binet’s Formula Formula
Binet’s formula gives the Fibonacci number at a term position without adding all previous Fibonacci numbers one by one.
F_n = (phi^n - psi^n)/sqrt(5)
- Fn = the Fibonacci number at position n
- n = the term number, starting with F0 = 0
- phi = the golden ratio, (1 + sqrt(5))/2, approximately 1.61803
- psi = (1 – sqrt(5))/2, approximately -0.61803
- sqrt(5) = the square root of 5, approximately 2.23607
To calculate a Fibonacci number, enter the term number n. The calculator applies Binet’s formula and rounds the result to the nearest whole number.
To find the term number from a Fibonacci number, enter Fn. The calculator checks term values until it finds a Fibonacci number that matches the value you entered.
Common Fibonacci Values
The table below shows common values you can use to check your result.
| Term n | Fibonacci number Fn |
|---|---|
| 0 | 0 |
| 1 | 1 |
| 2 | 1 |
| 5 | 5 |
| 10 | 55 |
| 15 | 610 |
| 20 | 6765 |
| 25 | 75025 |
| 30 | 832040 |
Example Calculations
Example 1: Find the Fibonacci number when n = 10
Use Binet’s formula:
F_10 = (phi^10 - psi^10)/sqrt(5)
Using phi ≈ 1.61803, psi ≈ -0.61803, and sqrt(5) ≈ 2.23607:
F10 = 55
Example 2: Find the term number for Fibonacci number 610
You need the value of n where:
610 = (phi^n - psi^n)/sqrt(5)
Checking Fibonacci values gives:
F15 = 610
So the term number is n = 15.
FAQ
What is Binet’s formula used for?
Binet’s formula is used to find a Fibonacci number directly from its term number. Instead of calculating each earlier term, you can enter n and calculate Fn using powers of the golden ratio.
Why does the result get rounded?
Fibonacci numbers are whole numbers, but Binet’s formula uses irrational numbers such as sqrt(5) and the golden ratio. Decimal rounding errors can appear during calculation, so the final value is rounded to the nearest integer.
Why might a Fibonacci number not return a term number?
If the number you enter is not actually in the Fibonacci sequence, there is no matching term number. For example, 21 returns n = 8, but 22 does not match any Fibonacci term.
