Question
Concentric circles of radii $$1, 2, 3, . . . .100 \,cm$$ 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 regions on $$sq\,cm$$ is equal to
A.
$$1000\,\pi $$
B.
$$5050\,\pi $$
C.
$$4950\,\pi $$
D.
$$5151\,\pi $$
Answer :
$$5050\,\pi $$
Solution :
$$\pi \left[ {\left( {{r_2}^2 - {r_1}^2} \right) + \left( {{r_4}^2 - {r_3}^2} \right) + ..... + \left( {{r_{100}}^2 - {r_{99}}^2} \right)} \right]$$
$$\eqalign{ & \therefore {r_2} - {r_1} = {r_4} - {r_3} = ..... \cr & = {r_{100}} - {r_{99}} = 1 \cr & = \pi \left[ {{r_1} + {r_2} + {r_3} + {r_4} + ..... + {r_{100}}} \right] \cr & = \pi \left[ {1 + 2 + 3 + ..... + 100} \right] \cr & = 5050\,\pi \,sq\,cm \cr} $$
$$\pi \left[ {\left( {{r_2}^2 - {r_1}^2} \right) + \left( {{r_4}^2 - {r_3}^2} \right) + ..... + \left( {{r_{100}}^2 - {r_{99}}^2} \right)} \right]$$
$$\eqalign{ & \therefore {r_2} - {r_1} = {r_4} - {r_3} = ..... \cr & = {r_{100}} - {r_{99}} = 1 \cr & = \pi \left[ {{r_1} + {r_2} + {r_3} + {r_4} + ..... + {r_{100}}} \right] \cr & = \pi \left[ {1 + 2 + 3 + ..... + 100} \right] \cr & = 5050\,\pi \,sq\,cm \cr} $$