A square is drawn by joining the mid points of the sides of a given square in the same way and this process continues indefinitely. If a side of the first square is 4 cm, determine the sum of the areas all the square.

A. 32 C$\mathrm{m}^{2}$ B. 16 C$\mathrm{m}^{2}$ C. 20 C$\mathrm{m}^{2}$ D. 64 C$\mathrm{m}^{2}$ Answer: Option A
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Solution(By Apex Team)

Side of the first square is 4 cm. side of second square = $2 \sqrt{2}$ cm. Side of third square = 2 cm. and so on, i.e. $4,2, \sqrt{2}, \sqrt{2}, 1 \ldots$ Thus, area of these square will be = $16,8,4,2,1, \frac{1}{2} \ldots$ Hence, Sum of the area of first, second, third square $\begin{aligned}&=16+8+4+2+1+\ldots\\ &=\frac{16}{1-\frac{1}{2}}\\ &=32\mathrm{~cm}^2\end{aligned}$

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