A square is drawn by joining mid point of the sides of a square. Another square is drawn inside the second square in the same way and the process is continued infinitely. If the side of the first square is 16 cm, then what is the sum of the areas of all the squares ?

Explanation

Let a be the side lengh of square, than
`AB=BC=CD=DA=a`
`therefore` E,F,G,H are the mid-points of AB,BC,CD and DA, respectively.
`therefore EF=FG=GH=HE=(a)/(sqrt(2))`
and I,J,K,L are the mid-points of EF,FG,GH and HE, respectively.
`therefore IJ=JK=KL=LI=(a)/(2)`
Similarly, `MN=NO=OP=PM=(a)/(2sqrt(2))` and `QR=RS=ST=TQ=(a)/(4),"....."`
S=Sum of areas =`ABCD+EFGH+IJKL+MNOP+QRST+"..."`
`a^(2)+((a)/(sqrt2))^(2)+((a)/(sqrt2))^(2)+((a)/(2sqrt2))^(2)+"...."`
`=a^(2)(1+(1)/(2)+(1)/(4)+(1)/(8)+"...")`
`=a^(2)((1)/(1-(1)/(2)))=2a^(2)=2(16)^(2) " " [therefore a=16" cm "]`
`=512sq " cm "`
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