<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 the starting value. For first n even natural numbers (2, 4, 6, ..., 2n), use the formula: Variance = (n²-1)/3 × d², where d is the common difference.
<p><strong>Step 1:</strong> First 50 even natural numbers are: 2, 4, 6, 8, ..., 100</p><p>This is an AP with first term a = 2, common difference d = 2, and n = 50 terms.</p><p><strong>Step 2:</strong> Mean x̄ = (first term + last term)/2 = (2 + 100)/2 = 51</p><p><strong>Step 3:</strong> For an arithmetic progression, variance σ² = d²(n² - 1)/12</p><p>σ² = 4 × (2500 - 1)/12 = 4 × 2499/12 = 9996/12 = 833</p><p><strong>Step 4:</strong> Alternatively, using σ² = (n² - 1)d²/12 where n = 50 and d = 2:</p><p>σ² = (2500 - 1) × 4/12 = 2499 × 4/12 = 833</p><p>∴ Answer: <strong>833</strong></p>
Correct Answer: D