Question

Concentric circles of radii $1,2,3,...,100cm$ are drawn. The interior of the smallest circle is coloured red and the angular regions are coloured alternately green and red, so that no two adjacent regions are of the same colour. The total area of the green region in sq.cm is equal to

• A. $1000\pi$
• B. $5050\pi$
• C. $4950\pi$
• D. $5151\pi$

Answer 1

The correct option is B $5050\pi$

Area of circle colored green$=\pi [2^2-1^2]$
$\therefore$ Angular regions of circle with radii 2,4,6,8,....100 have green color
$\therefore$ Total area of green colored regions
$=\pi [2^2-1^2]+\pi [4^2-3^3]+\pi [6^2-5^2]+......+\pi [{100}^2-{99}^2]$
$=\pi [(2-1)(2+1)+(4-3)(4+3)+(6-5)(6+5)+.......+(100-99)(100+99)]$
$=\pi [3+7+11+....+199]$
No. of terms in 3,7,11....199$=\frac{199-3}{4}+1$
$=\frac{196}{4}+1=50$
Total area $=\pi \times 50\times \left[\frac{3+199}{2}\right]$
$=\pi \times 50\times 101$
$=5050\pi$