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 $4\,cm$ and the sum of the area of all the squares is $\alpha $ then find the value of $\alpha /4.$

Answer 1

If a side of any square is $x\,cm,$ 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 $4^{2}=16$ $sq\,cm.$ The area of the second square is $8\,sq\,cm,$ the area of the third square is $4\,sq\,cm$ and so on. Hence the sum of the areas is given by $16+8+4+2+$ .... upto infinite
$=\frac{16}{1 - \left( 1 / 2 \right)}$
$=\frac{16}{1 / 2}$
$=16\times 2$
$=32\,cm^{2}Ans$