Question

Concentric circles of radius $1,2,3,\dots ,100$ units are drawn such that the interior of the smallest circle is coloured red and the angular regions are coloured alternatively green and red, so that no two adjacent regions are of the same colour. Then the area of green region (in sq. units) is

• A. $5000\pi$
• B. $5050\pi$
• C. $4950\pi$
• D. $4900\pi$

Answer 1

The correct option is B $5050\pi$


Area of all the green region $=\pi \left[\right(2^2-1^2)+(4^2-3^2)+\cdots +({100}^2-{99}^2\left)\right]$
$=\pi \left[\right(2+1\left)\right(2-1)+(4+3\left)\right(4-3)+(6+5\left)\right(6-5)+\cdots +(100+99\left)\right(100-99\left)\right]$
$=\pi [3+7+11+\cdots +199]$
$=\pi \left[\frac{50}{2}\right(3+199\left)\right]$
$=5050\pi$