TITLE: Why do ratios of these Fibonacci-type sequences approach $\pi$?
QUESTION [8 upvotes]: Define $A_n$ by $A_1=12$, $A_2=18$, and $A_n=A_{n-1}+A_{n-2}$ for $n\ge3$. Similarly define $B_n$ by $B_1=5$, $B_2=5$, and $B_n=B_{n-1}+B_{n-2}$ for $n\ge3$.
Terms of $A_n$: $12, 18, 30, 48, 78,\dots$
Terms of $B_n$: $5, 5, 10, 15, 25,\dots$
I found that dividing element $A_n$ by $B_n$ where $n$ approaches $\infty$ appears to result in:
$$\lim\limits_{n\to \infty}\left ( \frac{A_n}{B_n} \right ) = \pi$$
My question is, why does the ratio appear to converge towards $\pi$, and what is the significance of $5, 12, 18$ as to why this happens?

REPLY [4 votes]: Recurrences of the form
$$T_n=T_{n-1}+T_{n-2}$$ are linear and known to have a general solution of the form $$T_n=C_0z_0^n+C_1z_1^n,$$ where $z_0,z_1$ are the roots of the "characteristic equation", $z^2=z+1$.
By the usual formulas,
$$z_0,z_1=\frac{1\pm\sqrt5}2.$$
Using the initial conditions,
$$T_0=C_0+C_1,\\T_1=C_0z_0+C_1z_1.$$
As $|z_0|>|z_1|$, the first term quickly dominates and
$$T_n\approx\frac{T_1-z_1T_0}{z_0-z_1}z_0^n.$$
In your case, for large $n$,
$$\frac{A_n}{B_n}\approx\frac{A_1-z_1A_0}{B_1-z_1B_0}=\frac{3\sqrt5+9}5=3.1416407864999\cdots$$