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Binet's Formula — Definition, Formula & Examples

Binet's Formula is a closed-form expression that calculates the nnth Fibonacci number directly, without computing all the preceding terms. It uses the golden ratio ϕ\phi and produces exact integer results despite involving irrational numbers.

For integer n≥0n \geq 0, the nnth Fibonacci number FnF_n (with F0=0F_0 = 0, F1=1F_1 = 1) is given by Fn=ϕn−ψn5F_n = \dfrac{\phi^n - \psi^n}{\sqrt{5}}, where ϕ=1+52\phi = \dfrac{1+\sqrt{5}}{2} and ψ=1−52\psi = \dfrac{1-\sqrt{5}}{2} are the roots of the characteristic equation x2−x−1=0x^2 - x - 1 = 0.

Key Formula

Fn=ϕn−ψn5F_n = \frac{\phi^n - \psi^n}{\sqrt{5}}
Where:
  • FnF_n = The nth Fibonacci number (starting with F₀ = 0, F₁ = 1)
  • ϕ\phi = The golden ratio, (1 + √5)/2 ≈ 1.6180
  • ψ\psi = The conjugate of the golden ratio, (1 − √5)/2 ≈ −0.6180
  • nn = A non-negative integer index

How It Works

The Fibonacci sequence is defined recursively: each term is the sum of the two before it. Binet's Formula bypasses this recursion by expressing FnF_n as a function of nn alone. The key ingredients are ϕ≈1.618\phi \approx 1.618 (the golden ratio) and ψ≈−0.618\psi \approx -0.618. Since ∣ψ∣<1|\psi| < 1, the term ψn\psi^n shrinks rapidly toward zero as nn grows, so for large nn you can approximate Fn≈ϕn5F_n \approx \frac{\phi^n}{\sqrt{5}} rounded to the nearest integer.

Worked Example

Problem: Use Binet's Formula to find the 10th Fibonacci number, F₁₀.
Identify constants: Compute the two key values.
ϕ=1+52≈1.61803,ψ=1−52≈−0.61803\phi = \frac{1+\sqrt{5}}{2} \approx 1.61803, \quad \psi = \frac{1-\sqrt{5}}{2} \approx -0.61803
Substitute n = 10: Raise each constant to the 10th power and subtract.
ϕ10≈122.9919,ψ10≈0.0081\phi^{10} \approx 122.9919, \quad \psi^{10} \approx 0.0081
Divide by √5: Apply the formula to get the Fibonacci number.
F10=122.9919−0.00815=122.98392.2361≈55F_{10} = \frac{122.9919 - 0.0081}{\sqrt{5}} = \frac{122.9839}{2.2361} \approx 55
Answer: F10=55F_{10} = 55, which matches the Fibonacci sequence: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55.

Why It Matters

Binet's Formula appears in discrete mathematics and linear algebra courses when studying linear recurrence relations. It demonstrates a powerful technique: solving a recursive sequence by finding roots of its characteristic equation, a method that extends to any linear recurrence with constant coefficients.

Common Mistakes

Mistake: Assuming the formula can't produce exact integers because it contains √5.
Correction: The irrational parts from ϕⁿ and ψⁿ always cancel exactly. The result is always a whole number for integer n ≥ 0.