<p>If the mean deviation of the numbers \(1, 1+d, \ldots, 1+100d\) from their mean is 255, then a value of \(d\) is</p>
Step-by-Step Solution
Key Concept: The mean of an arithmetic sequence equals the average of first and last terms, and mean deviation involves summing absolute deviations symmetrically distributed around the mean.
<p><strong>Step 1:</strong> Identify the sequence. We have 101 terms: 1, 1+d, 1+2d, ..., 1+100d (arithmetic sequence with first term a=1 and common difference d).</p><p><strong>Step 2:</strong> Find the mean. Mean = (First term + Last term)/2 = (1 + 1+100d)/2 = 1 + 50d</p><p><strong>Step 3:</strong> Calculate deviations from mean. The k-th term is 1+kd where k = 0,1,2,...,100. Deviation from mean = (1+kd) - (1+50d) = (k-50)d</p><p><strong>Step 4:</strong> Find mean deviation. Due to symmetry about k=50, we have pairs of deviations: ±d, ±2d, ±3d, ..., ±50d (and 0 at the middle).</p><p><strong>Step 5:</strong> Sum of absolute deviations = |d|·1 + |d|·2 + |d|·3 + ... + |d|·50 + 0 + |d|·50 + ... + |d|·2 + |d|·1 = 2|d|(1+2+3+...+50) = 2|d| · (50·51/2) = 2|d| · 1275</p><p><strong>Step 6:</strong> Mean deviation = (Sum of absolute deviations)/101 = (2|d|·1275)/101 = (2550|d|)/101 = 255</p><p><strong>Step 7:</strong> Solve for d. 2550|d| = 255 × 101 = 25755, so |d| = 25755/2550 = 10.1</p><p><strong>Step 8:</strong> Simplify. 25755/2550 = 5151/510 = 1701/170 = 10.1, or more precisely |d| = 10.1. Since d can be positive or negative, <strong>d = ±10.1 or d = ±51/5</strong></p><p>∴ Answer: D</p>
Correct Answer: D