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 the total number of stones be 2n + 1

∴  There are n stones on each side of the middle stone. Let the man starts collecting stones from the extreme left stone.  The distance covered by man to bring the extreme left stone to the middle stone = 10n metres. The distance covered by man to bring the (n - 1)th stone on left to the middle stone = 2 10 (n - 1) metres.

∴   The total distance covered by the man to bring all the stones from left to the middle stone 

                = 10n + 20(n - 1) + 20(n - 2) + ......... + 20

Similarly, the total distance covered by the man to bring all the stones from the right to the middle stone 

                                               = 20 + 20 (n - 1) + ......... + 20

∴     Total distance covered by the man to bring all the stones to the middle stone

                                              = 30n + 40 (n - 1) + 40 (n - 2) +........ + 40 (1)

                                            = 40 [n + (n - 1) + (n - 2) + ....... + 1] - 10n = 

                                           

But the distance covered is 3 km or 3000 metres.

∴    

               

                         

                             n = 12

∴      Number of stones = 2(12) + 1 = 25

Here,     a = 3,00,000,  d = 10,000, n = 20


∴                            

                                     = 10 (7,90,000) = 79,00,000.