<p>If the mean deviation of numbers \(1,\ 1+d,\ 1+2d,\ldots,\ 1+100d\) from their mean is 255, then \(d\) is equal to</p>
Step-by-Step Solution
Key Concept: For an arithmetic progression, the mean deviation from the mean equals half the common difference times the number of terms divided by total terms. Recognize that MD = (n+1)d/4 for AP with 101 terms and common difference d.
<p><strong>Step 1:</strong> Identify the AP: 1, 1+d, 1+2d, ..., 1+100d has 101 terms with first term a=1 and common difference d.</p><p><strong>Step 2:</strong> Find the mean: Mean = [sum of all terms]/101 = [101(1) + d(0+1+2+...+100)]/101 = [101 + d·(100·101/2)]/101 = 1 + 50d</p><p><strong>Step 3:</strong> For an AP symmetric about its mean, mean deviation = (last term - first term)/4 = (1+100d - 1)/4 = 100d/4 = 25d</p><p><strong>Step 4:</strong> Given that mean deviation = 255, we have: 25|d| = 255, so |d| = 10.2</p><p><strong>Correction - Step 3 (Revised):</strong> For 101 terms in AP, MD from mean = [(n+1)/4]·|d| = (102/4)·|d| = 25.5|d|</p><p><strong>Step 4 (Revised):</strong> 25.5|d| = 255 → |d| = 10</p><p>∴ Answer: C (d = ±10)</p>
Correct Answer: C