Question

Let $S_1$ be a square of side $5$ cm. Another square $S_2$ is drawn by joining the mid-points of the sides of $S_1$. Square $S_3$ is drawn by joining the mid-points of the sides of $S_2$ and so on. Then, (area of $S_1+$ area of $S_2+$ area of $S_3+\cdots +$ area of $S_{10}$) is

• A. $25(1-\frac{1}{2^{10}})$ sq. cm
• B. $50(1-\frac{1}{2^{10}})$ sq. cm
• C. $2(1-\frac{1}{2^{10}})$ sq. cm
• D. $1-\frac{1}{2^{10}}$ sq. cm
• E. $10(1-\frac{1}{2^{10}})$ sq. cm

Answer 1

The correct option is B $50(1-\frac{1}{2^{10}})$ sq. cm

Area of $S_1=25$ sq. cm
Area of $S_2=\frac{25}{2}$ sq. cm
Area of $S_3=\frac{25}{4}$ sq. cm and so on.
Total area $=25+\frac{25}{2}+\frac{25}{4}+\cdots$ upto $10$ terms
$=25[1+\frac{1}{2}+\frac{1}{4}+\cdots ]$
$=25\left[\frac{1-{\left(\frac{1}{2}\right)}^{10}}{1-\frac{1}{2}}\right]$
$=50(1-\frac{1}{2^{10}})$ sq. cm