Question 33
In the following diagram, ABCD is a square with AB=10 cm. Let $A_1,B_1,C_1,D_1$ be points on the sides of ABCD such that $AA_1=BB_1=CC_1=DD_1=\frac 16 AB$. Similarly, let $A_2,B_2,C_2,D_2$ be points on the sides of $A_1 B_1 C_1 D_1$ such that $A_1 A_2=B_1 B_2=C_1 C_2=D_1 D_2=\frac 16 A_1 B_1$. Repeat this procedure to construct infinitely many squares $$ABCD,\ A_1 B_1 C_1 D_1,\ A_2 B_2 C_2 D_2,\ldots.$$ Find the sum of their areas in cm².
