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 adjecent regions are of the same colour. Then find the total area of green regions in sq.cm.

Answer 1

Smallest circle is colured Red

Then Area of Green region for 1st two Circles$=\pi \times 2^2-\pi \times 1^2$

$=\pi \times 1\times 3$

Then Area of next green region $=\pi \times 4^2-\pi \times 3^2=\pi (4-3)(4+3)=7\pi$

and so on

Last green region area $=\pi \times {100}^2-\pi \times {99}^2=\pi (100-99)(100+99)=199\pi$

Total Green area

$=\pi \times 1\times 3+\pi \times 1\times 7+\pi \times 1\times 11+...+\pi \times 1\times 199$

$=\pi (3+7+11+...+199)$

$a=3,n=50,d=4$

$S=\left(\frac{50}{2}\right)(3+199)=50\times 101$

Area of green region $=50\times 101\pi {cm}^2=5050\pi {cm}^2=15871.4{cm}^2$