Read the equation
The closed form combines the two roots of r²=r+1: φ and −1/φ. Subtracting their nth powers and dividing by √5 produces the standard Fibonacci integer Fₙ. It explains the ratio limit, although direct floating-point evaluation can lose exact integer accuracy for large n.
Terms & definitions
| Term | Meaning in this source |
|---|---|
| F_n | nth Fibonacci integer |
| golden | Golden ratio φ |
| n | Nonnegative integer index |
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Original context
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We provide additional detail for Theorem~\ref{thm:convergence}.
\textbf{Fibonacci Ratio Convergence.} The ratio $\fib{n}/\fib{n-1}$ converges to $\golden$ exponentially fast. Using Binet's formula:
\begin{equation}
\fib{n} = \frac{\golden^n - (-\golden)^{-n}}{\sqrt{5}}
\end{equation}
For large $n$, $(-\golden)^{-n} \to 0$, so:
\begin{equation}
\frac{\fib{n}}{\fib{n-1}} \approx \frac{\golden^n}{\golden^{n-1}} = \golden
\end{equation}Exact original notation
\fib{n} = \frac{\golden^n - (-\golden)^{-n}}{\sqrt{5}}