Permutations and Combinations

Question

# A square is drawn by joining the mid-points of the sides of a given square. A third square is drawn inside the second square in the same way, and the process continues indefinitely. If a side of the first square is 16 cm., determine the sum of the areas of all the squares.

Let ABCD be the given square of side 16cm as shown in the figure. Let EFGH be the second square drawn by joining the mid-points of the sides of given square ABCD.
BF = EB = 8

Area of first square =
Area of 2nd square =
Now, side of the third square =
∴  Area of third square =  sq. cm. Since the number of squares is infinite
∴  Sum of areas of all such squares = 256 + 128 + 64 + .......