Question

A square is drawn by joining the mid points of the sides of a square. A third square is drawn inside the second square in the same way and the process is continued infinitely. If the side of the square is $10$ cm, then the sum of areas of all the squares so formed is

• A. $100$ cm${}^2$
• B. $200$ cm${}^2$
• C. $250$ cm${}^2$
• D. None of these

Answer 1

The correct option is B $200$ cm${}^2$

Side of the square is $10cm$
So, area of square $=100c{m}^2$
Now Area of second Square is $=\frac{1}{2}\times$ area of first square$=50c{m}^2$
Area of third square $=25c{m}^2$
Clearly $100,50,25...$ a G.P
With first term $\left(a\right)=100$
and common ration $\left(r\right)=\frac{1}{2}$
So, Sum of all square = Sum of G.P to infinity $=\frac{a}{1-r}=\frac{100}{1-\frac{1}{2}}=200c{m}^2$