LaTeX
F_n=\frac{\varphi^n-(1-\varphi)^n}{\sqrt5}Variables
- F_n: nth Fibonacci number
- n: nonnegative integer
- φ: (1+√5)/2
How to use this formula
Expresses the nth Fibonacci number using powers of the golden ratio.
Important notes
- Floating-point evaluation should be rounded carefully for large n.
- Indexing conventions may start with F₀=0 or F₁=1.
Quick example
Using n=10 gives F₁₀=55.
Applicability, worked calculation, and verification
Assumptions and domain checks
- Floating-point evaluation should be rounded carefully for large n.
- For the Binet Formula for Fibonacci Numbers, every denominator must be nonzero, and the numerator and denominator must remain correctly grouped.
- For a real-valued result, each even-root radicand must be nonnegative unless complex values are intended.
- Confirm the finite-set, indexing, ordering, and counting conventions assumed by the expression.
Worked example
Using n=10 gives F₁₀=55.
Common mistakes
- When copying Binet Formula for Fibonacci Numbers, keep the complete numerator and denominator grouped; a missing brace or parenthesis changes the result.
- Verify the result of Binet Formula for Fibonacci Numbers with a known case, inverse operation, dimensional check, or independent calculation before publishing it.
Continue the workflow
Use Binet Formula for Fibonacci Numbers in your own work
- Check the domainMatch the variables and assumptions to the problem before substituting values.
- Copy the exact notationPreserve grouping, signs, and exponents in
F_n=\frac{\varphi^n-(1-\varphi)^n}{\sqrt5}. - Edit or convertOpen the expression in the LaTeX editor, then export it for your document or web page.
Review and verification
Last reviewed: 2026-07-23
Automated quality check: Kept noindex until the missing evidence is supplied.
Formula references
- Digital Library of Mathematical FunctionsNational Institute of Standards and Technology — Definitions, notation, identities, and reference material for mathematical functions.
Frequently asked questions
What is the Binet Formula for Fibonacci Numbers used for?
Expresses the nth Fibonacci number using powers of the golden ratio.
Can I copy this formula as LaTeX?
Yes. Copy F_n=\frac{\varphi^n-(1-\varphi)^n}{\sqrt5} or open it in the LaTeX editor.
What should I check before using it?
Confirm that each variable, unit, domain restriction, and assumption matches the problem.