A square is drawn by joining the mid points of the sides of a given square in the same way and this process continues indefinitely. If a side of the first square is 4 cm, determine the sum of the areas all the square.

Solution (By Examveda Team)

Side of the first square is 4 cm.
side of second square = $$2\sqrt 2 {\kern 1pt} $$ cm.
Side of third square = 2 cm.
and so on, i.e. 4, 2, $$\sqrt 2 $$ , $$\sqrt 2 $$ , 1 ........
Thus, area of these square will be = 16, 8, 4, 2, 1, $$\frac{1}{2}$$ ..........
Hence, Sum of the area of first, second, third square
$$\eqalign{ & = 16 + 8 + 4 + 2 + 1 + {\kern 1pt} \,...... \cr & = {\frac{{16}}{{ {1 - {\frac{1}{2}} } }}} \cr & = 32\,{\kern 1pt} c{m^2} \cr} $$