Question

The length of the side of square is ${}^{\prime }{a}^{\prime }$ meter. A second square is formed by joining the middle points of the side of the squares.Then a third square is formed by joining the middle points of the side of the second square and so on .Then the sum of the area of squares which carried upto infinity is

• A. $a^2$
• B. $2a^2$
• C. $3a^2$
• D. $4a^2$

Answer 1

Answer: (B)

$2a^2$

the area of square of side a is $a^2$
the side of the square formed by joining the midpoints of original square is $\frac{a}{\sqrt{2}}$
so the area of the second square is $\frac{a^2}{2}$
similarly, area of the third square formed the same way is $\frac{a^2}{4}$
the sum of areas of all those squares is an infinite geometric series with first term as $a^2$ and common ratio as $1/2$
so the sum is equal to $\frac{firstterm}{1-commonratio}=2a^2$