<p>Concentric circles of radii 1, 2, 3, …, 100 cm are drawn. The interior of the smallest circle is colored red and the angular regions are colored alternately green and red, so that no two adjacent regions are of the same color. Then, the total area of the green regions in sq. cm is equal to</p>
Step-by-Step Solution
Key Concept: Recognize that each annular region (ring) between consecutive circles forms an alternating color pattern. Calculate the area of each ring and sum only the green regions using the difference of consecutive squares formula.
<p><strong>Step 1:</strong> Identify the coloring pattern. The smallest circle (radius 1) is <strong>red</strong>. The annular regions alternate: the first ring (between r=1 and r=2) is <strong>green</strong>, the second ring (between r=2 and r=3) is <strong>red</strong>, and so on.</p><p><strong>Step 2:</strong> Area of the nth annular region (between radius n and n+1) = π(n+1)² - πn² = π(2n+1)</p><p><strong>Step 3:</strong> Green regions are at positions 1, 3, 5, 7, ... (odd-indexed rings). These correspond to n = 1, 3, 5, ..., 99.</p><p><strong>Step 4:</strong> Total green area = π[2(1)+1] + π[2(3)+1] + π[2(5)+1] + ... + π[2(99)+1]</p><p>= π[3 + 7 + 11 + 15 + ... + 199]</p><p><strong>Step 5:</strong> This is an arithmetic series with first term a = 3, common difference d = 4, and 50 terms (n = 1, 3, 5, ..., 99 gives 50 odd numbers).</p><p>Sum = (50/2)[2(3) + 49(4)] = 25[6 + 196] = 25 × 202 = 5050</p><p>∴ Total green area = 5050π sq. cm</p><p><strong>Answer: C (5050π or equivalent form)</strong></p>
Correct Answer: C