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 AShow Answer
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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