Question 85
As shown in the diagram, there is a sequence of the curves $P_0, P_1, P_2, \dots$. It is known that the region enclosed by $P_0$ has area $1$ and $P_0$ is an equilateral triangle. We obtain $P_{k+1}$ from $P_k$ by operating as follows: Trisecting every side of $P_k$, then we construct an equilateral triangle outwardly on every side of $P_k$ sitting on the middle segment of the side and finally remove this middle segment ($k = 0, 1, 2, \dots$). Write $S_n$ as the area of the region enclosed by $P_n$.
(1) Find a formula for the general term of the sequence of numbers $\{S_n\}$;
(2) Find $\lim_{n \to \infty} S_n$.
$P_0$
$P_1$
$P_2$
...
(1) Find a formula for the general term of the sequence of numbers $\{S_n\}$;
(2) Find $\lim_{n \to \infty} S_n$.
$P_0$
$P_1$
$P_2$
...