Question

The length of a side of a square is $a$ meters. A second square is formed by joining the middle points of this square. Then a third square is formed by joining the middle points of the second square and so on. The process is carried on ad-infinitum. Find the sum of the areas of the squares.

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

Answer 1

The correct option is C $2a^2.$

Sides are $a,\frac{a}{\sqrt{2}},\frac{a}{2},\frac{a}{2\sqrt{2}}....$
$\therefore$ Areas are $a^2,\frac{a^2}{2},\frac{a^2}{4},\frac{a^2}{8}....$
Sum of the areas = $\frac{a^2}{(1-\frac{1}{2})}$ ($\because S_{\infty }=\frac{a}{1-r}$)
$\therefore$ Required sum is $2a^2$
Hence, option C.