Enter the Nth term into the calculator to determine the corresponding Fibonacci number using Binet's formula.
Binet's Formula
The following formula is used to calculate the Nth Fibonacci number.
F<sub>n</sub> = (φ<sup>n</sup> - ψ<sup>n</sup>) / √5
Variables:
- Fn is the Nth Fibonacci number
- φ (phi) is the golden ratio, approximately 1.61803398875
- ψ (psi) is the conjugate of the golden ratio, approximately -0.61803398875
- n is the Nth term in the Fibonacci sequence
To calculate the Nth Fibonacci number using Binet's formula, raise φ to the power of n, raise ψ to the power of n, subtract the second result from the first, and divide by the square root of 5. The result is rounded to the nearest whole number, as Fibonacci numbers are integers.
What is Binet's Formula?
Binet's formula is an explicit formula used to find the Nth term of the Fibonacci sequence without having to calculate all the preceding terms. It is derived from the closed-form solution of the Fibonacci sequence and involves the golden ratio, φ, and its conjugate, ψ. This formula allows for the rapid calculation of large Fibonacci numbers.
How to Calculate the Nth Fibonacci Number?
The following steps outline how to calculate the Nth Fibonacci Number using Binet's Formula.
- First, determine the Nth term (n) you want to calculate for the Fibonacci sequence.
- Next, use the formula Fn = (φn - ψn) / √5, where φ is the golden ratio and ψ is its conjugate.
- Finally, calculate the Nth Fibonacci number (Fn).
- After inserting the variable and calculating the result, check your answer with the calculator above.
Example Problem :
Use the following variable as an example problem to test your knowledge.
Nth term (n) = 10
