The sum of areas of triangles omitted infinitely many times within an equilateral triangle ABC 
Triangle $ABC$ is an equilateral triangle. Four  new equilateral triangles are  formed by joining the midpoints of the sides $A, B, C$ in $\triangle ABC$ Triangle, with the white triangle in İmage 1, and it is taken out. 
Each triangle now formed in $\triangle ABC$ has an area of $\frac 14$ of the area of $\triangle ABC$, as seen in Image 1.
The same process of joining the midpoints in each triangle and taking out the newly formed white triangles is repeated infinitely many times as exemplified in Image 2 and Image 3.
If $\triangle ABC$ has an area of $1\, m^2,\,$ what is the sum of the areas of white triangles taken out? 
I know the answer, which I've posted below.

$$\begin{align} S &=\sum_{n=1}^\infty a\cdot r^{n-1}\\ \\ & = \frac{1}4+\frac{3}{16}+\frac{9}{64}+\dots \\ \\ 
                         &= \frac{1}4\cdot\frac{3}{4}^{0}+\frac{1}4\cdot\frac{3}{4}^{1}+\frac{1}4\cdot\frac{3}{4}^{2}+\dots \\ \\ \end{align}$$
$$ a=\frac{1}4, r=\frac{3}4\implies S=  1  $$
