Solutions - 0150-07

Compute the first few areas, with $a_0$ being the area of the largest triangle:

$$\begin{array}{cc}{a}_0& =\frac{1}{2}\cdot \frac{1}{2}\cdot \frac{1}{2}=\frac{1}{8}\\ & \\ a_1& =\frac{1}{2}\cdot \frac{1}{2}\cdot \frac{1}{4}=\frac{1}{16}\\ & \\ a_2& =\frac{1}{2}\cdot \frac{1}{2}\cdot \frac{1}{8}=\frac{1}{32}\end{array}$$

This is a geometric series with $r=\frac{1}{2}$:

$$A=\sum _{n=0}^{\infty }\frac{1}{8}{\left(\frac{1}{2}\right)}^n$$

Geometric series total sum formula:

$$S=\frac{a_0}{1-r}\gg \gg \frac{1/8}{1-1/2}\gg \gg \frac{1}{4}$$