Question

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 this process continues indefinitely. If a side of the first square is $4cm$ and the sum of the area of all the squares is $\alpha$ then find the value of $\alpha /4.$

Answer 1

Answer: 8

If a side of any square is $xcm,$ then the side of the square obtained by joining its mid-points is given by

$\sqrt{{\left(\frac{x}{2}\right)}^2+{\left(\frac{x}{2}\right)}^2}=\frac{x}{\sqrt{2}}cm$
and such its area is ${\left(\frac{x}{\sqrt{2}}\right)}^2=\frac{x^2}{2}{\left(cm\right)}^2$
Now the area of the first square is $4^2=16$ $sqcm.$ The area of the second square is $8sqcm,$ the area of the third square is $4sqcm$ and so on. Hence the sum of the areas is given by $16+8+4+2+$ .... upto infinite
$=\frac{16}{1-\left(1/2\right)}$
$=\frac{16}{1/2}$
$=16\times 2$
$=32c{m}^{2}Ans$