Q:

Problem with shaded regions in a square

The sides of a square are 16 cm in length. The midpoints of the sides of this square are joined to form a new square and four triangles (diagram 1). The process is repeated twice as shown in the diagrams 2 and 3.

Consider an initial square of side length $$
Kcm\,.
$$ The process described above is repeated indefinitely. The total area of shaded regions is
$$
Kcm^{2}\,.
$$
What is the value of K?

A:

An initial square of side-length $k$ cm has an area of $k^2$ cm, and you can see that $A_1=k^2/8$. Looking at the diagram you can see that $A_{n+1}=A_n/2$. This is a geometric series with $a=k^2/8$ and $r=1/2$. So the sum of all the areas is $$\frac{k^2/8}{1-1/2}=\frac{k^2}{4}$$

