Successive Fibonacci ratios approach φ

Read the equation

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

How it works

Read the definition from its right-hand expression to the quantity on the left. Its surrounding source records the intended input, state, or parameter context. Where the source leaves an operation unspecified, this page preserves that gap.

Assumptions & limits

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

Original context

\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}

The error is $O(\golden^{-2n})$, converging exponentially.

\textbf{Equilibrium Analysis.} At equilibrium, all time derivatives vanish. From Equation~\ref{eq:dC}:
\begin{equation}
0 = \alpha(E^* - C^*) + \beta L^* \langle \sin(\golden t) \rangle \cdot \golden

\frac{\fib{n}}{\fib{n-1}} \approx \frac{\golden^n}{\golden^{n-1}} = \golden

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