Calculate any Fibonacci number with Binet’s formula, step by step, or find the term number n for a Fibonacci number or the nearest one to it.
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Binet’s Formula
Binet’s formula gives the Fibonacci number at a term position without adding all previous Fibonacci numbers one by one.
- Fn = the Fibonacci number at position n
- n = the term number, starting with F0 = 0
- phi = the golden ratio, (1 + √(5))/2, approximately 1.61803
- psi = (1 - √(5))/2, approximately -0.61803
- √(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:
Using phi ≈ 1.61803, psi ≈ -0.61803, and √(5) ≈ 2.23607:
F10 = 55
Example 2: Find the term number for Fibonacci number 610
You need the value of n where:
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 √(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.
