MathDB

There is a family of curves:

P_0,P_1,P_2,\cdots

P_0

is a regular triangle, whose area is

1

. For all

k\in\mathbb{Z}_+

P_k

is defined in this way: trisect all sides of

P_{k-1}

, and draw outward a regular triangle with side of the segment in the middle, then cut off the segment in the middle.

S_n

is the area of

P_n

. (a) Find

S_n

. (b) Find

\lim_{n\to\infty}S_n

Problem Statement