Successive Fibonacci ratios approach φ

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For the standard Fibonacci sequence with F₀=0 and F₁=1, the ratio Fₙ/Fₙ₋₁ approaches the golden ratio as n grows. The contribution from the smaller-magnitude root of the recurrence vanishes relative to φⁿ. This theorem applies to the integer sequence; it does not directly prove convergence of a separately forced ODE that happens to use the ratio.

Terms & definitions

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Assumptions & limits

• The ratio requires a nonzero denominator; with the stated seeds use n≥2.

Original context

\end{definition}

The ratio of consecutive Fibonacci numbers converges to $\golden$:
\begin{equation}
\lim_{n \to \infty} \frac{\fib{n}}{\fib{n-1}} = \golden
\end{equation}

We also employ the golden angle:
\begin{equation}
\theta_g = \frac{2\pi}{\golden^2} \approx 137.508°
\end{equation}

\lim_{n \to \infty} \frac{\fib{n}}{\fib{n-1}} = \golden

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