Question

A square is drawn by joining the midpoint of the sides of square of sides 4cm. A third square is drawn inside second square in same way the process continued indefinitely. Determine the sum of areas of all the squares.

Answer 1

aea of first square =(4)^2
area of 2nd square =(2root2)^2
area of 3rd square =(2)^2
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now sum of area =(4)^2 + (2root2)^2 + (2)^2 + ........
=(4)^2 + (4)^2/2 + (4)^2/4 + (4)^2/16 + .....
=16 + 8 + 4 + 1 + ........
here we see sum of area of square is in GP which first term 16 and ratio constant=1/2
we know sum of infinity terms in GP =a/(1-r)
hence sum of area =32 (1-1/2)
=32