Closed Form of Fibonacci (Binet)

Key idea

Explicit formula using the golden ratio.

In depth

Fₙ = (φⁿ − ψⁿ)/√5 where φ = (1 + √5)/2 and ψ = (1 − √5)/2. It comes from the characteristic equation r² = r + 1. Since |ψ| < 1, Fₙ is the nearest integer to φⁿ/√5.

Worked example

F₁₀ = (φ¹⁰ − ψ¹⁰)/√5 = 55.

Formula

Real-world uses

• Calculating very large Fibonacci numbers instantly without adding all earlier terms

Common mistake

Expecting the formula with irrational numbers to give a non-integer; the irrational parts cancel.

How to learn this

1. Read the key idea and say it in your own words.
2. Work through the worked example, then try a similar problem.
3. Use the interactive lesson’s visual and mini-quiz to check the pattern.

Frequently asked questions

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Related expert topics in Discrete Math & Number Theory

• Strong Induction – Assume all earlier cases to prove the next.
• Well-Ordering Principle – Every nonempty set of positive integers has a least element.
• Partial Orders – Reflexive, antisymmetric, transitive relations.
• Cardinality of Sets – Comparing sizes of infinite sets.
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• Cantor’s Diagonal Argument – Proof that real numbers are uncountable.