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.

Explanation:-

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