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A square is formed with  side of 4 units. Another square is formed with the midpoints of the sides of the square. The process is repeated infinite times. The total area of the square is _________________________square units

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Firstly , area of outermost square  =  (4)2    =   16

           Then we need to find side length of next inner square..As it is given mid points are joined hence by Pythagoras theorem :

          side  = sqrt(22 + 22)   =  sqrt(8)

Hence area of next inner square  =  side * side  =  8

Likewise area of next to next inner square will come out to be 4 and this will continue infinitely..Hence we get the series : 

16 , 8 , 4 , 2 , ............... which is a decreasing G.P. with common ratio = 1/2 and initial term  =  16..

We know ,

Sum of infinite G.P.    =    a / (1 - r) where |r|  < 1 and hence a decreasing G.P. 

                                =    16 / (1 - 1/2)

                                =    32

Hence total area        =    32

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