Suppose that there is an equilateral triangle of area 10, call this $A_0$, we then trisect each side of this triangle, then cut off the corners, giving the polygon $A_1$, what is the area of this polygon, we then do this again to get $A_2$ (trisect sides, and cut corners), what is the area of this polygon? What is the area of $A_\infty$.
With this problem, I initially tried to generalise side length for $A_n$, then used the area of a polygon formula to find the area, I was wondering if anyone could come up with any other ideas, or alternatively do it the same way I did, to compare answers. Best of luck!
The answer for $A_n$ as n goes to infinity is $0$
Note that $$A_0 = 10$$
$$A_1 = 10(2/3)$$ $$A_2= 10(2/3)^2$$ $$A_3= 10(2/3)^3$$ $$.$$ $$.$$ $$.$$ $$A_n= 10(2/3)^{n}$$ $$ A_{\infty } =lim _{n\to \infty} 10(2/3)^n = 0 $$