Question

Let $S_{1}$ be a square of side $5\,cm$. Another square $S_{2}$ is drawn by joining the midpoints of the sides of $S_{1}$ Square $S_{3}$ is drawn by joining the midpoints of the sides of $S_{2}$ and so on. Then (area of $S_{1}$ + area of $S_{2}$ + area of $S_{3}$ $+\ldots+$ area of $S_{10}$) =

• A. $25\left(1-\frac{1}{2^{10}}\right) Cm^{2}$
• B. $50\left(1-\frac{1}{2^{10}}\right)Cm^{2}$
• C. $2\left(1-\frac{1}{2^{10}}\right)Cm^{2}$
• D. $1-\frac{1}{2^{10}}Cm^{2}$

Answer 1

Answer: (B)

$50\left(1-\frac{1}{2^{10}}\right)Cm^{2}$

Answer 2

Area of $S_{1}=25 cm ^{2}$
Area of $S_{2}=\frac{25}{2}\,cm ^{2}$
Area of $S_{3}=\frac{25}{4} \,cm ^{2}$ and so on
Total area $=25+\frac{25}{2}+\frac{25}{4}+\ldots$ upto 10 terms
$=25\left[1+\frac{1}{2}+\frac{1}{4}+\ldots\right]$
$=25\left[\frac{1-\left(\frac{1}{2}\right)^{10}}{1-\frac{1}{2}}\right]$
$=50\left(1-\frac{1}{2^{10}}\right) cm ^{2}$