<p>The variance of first 50 even natural numbers is</p>
Step-by-Step Solution
Key Concept: The variance of an arithmetic sequence depends only on the number of terms and common difference, not on the starting value. For first n even natural numbers (2, 4, 6, ..., 2n), use the formula Var = (d²/12)(n² - 1) where d is the common difference.
<p><strong>Step 1:</strong> Identify the sequence: First 50 even natural numbers are 2, 4, 6, ..., 100</p><p><strong>Step 2:</strong> This is an arithmetic progression with first term a = 2, common difference d = 2, and n = 50 terms</p><p><strong>Step 3:</strong> Mean = (first term + last term)/2 = (2 + 100)/2 = 51</p><p><strong>Step 4:</strong> For arithmetic progression, variance = d²(n² - 1)/12 where d is common difference</p><p><strong>Step 5:</strong> Var = 4(2500 - 1)/12 = 4(2499)/12 = 9996/12 = 833</p><p>Alternatively: Var = [(n² - 1)/12] × d² = [(2500 - 1)/12] × 4 = (2499/12) × 4 = 833</p><p><strong>∴ Answer: B (Variance = 833)</strong></p>
Correct Answer: B