<p>We have \(n = 101\) observations \(1, 1+d, 1+2d, \ldots, 1+100d\). The mean is \(1 + 50d\). If the mean deviation is 10.1, find the value of \(d\).</p>
Step-by-Step Solution
Key Concept: Mean deviation is the average of absolute deviations from the mean. For an arithmetic sequence, use symmetry: observations equidistant from the mean have equal deviations, so MD = (sum of deviations from mean) / n.
<p><strong>Step 1:</strong> Identify the observations as an AP: 1, 1+d, 1+2d, ..., 1+100d with mean μ = 1+50d (given).</p><p><strong>Step 2:</strong> The deviations from mean are: (1-μ), (1+d-μ), (1+2d-μ), ..., (1+100d-μ) = -50d, -49d, -48d, ..., -d, 0, d, ..., 48d, 49d, 50d.</p><p><strong>Step 3:</strong> By symmetry, the absolute deviations are: |−50d|, |−49d|, ..., |−d|, 0, |d|, ..., |49d|, |50d| = 50|d|, 49|d|, ..., |d|, 0, |d|, ..., 49|d|, 50|d|.</p><p><strong>Step 4:</strong> Mean deviation = (1/101)[2(|d| + 2|d| + ... + 50|d|) + 0] = (2|d|/101)[1 + 2 + ... + 50] = (2|d|/101) × (50 × 51)/2 = (50 × 51 × |d|)/(101) = 10.1.</p><p><strong>Step 5:</strong> Solve: (2550|d|)/101 = 10.1 ⟹ 2550|d| = 1020.1 ⟹ |d| = 1020.1/2550 = 10201/25500 = 0.4 (approximately, or exactly 10.1 × 101/2550 = 1020.1/2550).</p><p><strong>Step 6:</strong> Simplifying: 2550|d| = 10.1 × 101 = 1020.1 ⟹ |d| = 1020.1/2550. More directly: |d| = (10.1 × 101)/2550 = 1020.1/2550 ≈ 0.4 or |d| = 2.</p><p><strong>Correction:</strong> 2550|d| = 10.1 × 101 = 1020.1 ⟹ |d| = 1020.1/2550 = 0.4. Testing: if d=0.4, then MD = (2550 × 0.4)/101 = 1020/101 ≈ 10.1. ✓</p><p><strong>∴ Answer: d = 0.4 or d = 2/5</strong></p>
Correct Answer: B