<p>If the mean deviation about the median of the numbers \(a\), \(2a\), ..., \(50a\) is 50, then \(|a|\) equals</p>
Step-by-Step Solution
Key Concept: Mean deviation about median equals the average absolute deviation from the median. For an arithmetic sequence, the median is the average of middle terms, and by symmetry, MD = (sum of deviations from median)/(total terms).
<p><strong>Step 1:</strong> The sequence is a, 2a, 3a, ..., 50a (50 terms in arithmetic progression).</p><p><strong>Step 2:</strong> Since there are 50 terms (even), the median is the average of 25th and 26th terms:<br>Median = (25a + 26a)/2 = 51a/2</p><p><strong>Step 3:</strong> Mean deviation about median = (1/50)Σ|x_i - 51a/2|<br>= (1/50)Σ|ia - 51a/2| where i = 1 to 50<br>= (|a|/50)Σ|i - 25.5|</p><p><strong>Step 4:</strong> Calculate Σ|i - 25.5| for i = 1 to 50:<br>= |1-25.5| + |2-25.5| + ... + |25-25.5| + |26-25.5| + ... + |50-25.5|<br>= 24.5 + 23.5 + ... + 0.5 + 0.5 + 1.5 + ... + 24.5<br>= 2(0.5 + 1.5 + 2.5 + ... + 24.5)<br>= 2 × [0.5(1 + 2 + ... + 49)]/1<br>= 2 × (0.5 × 49 × 50/2) = 1225</p><p><strong>Step 5:</strong> Mean deviation = (|a|/50) × 1225 = 24.5|a| = 50<br>∴ |a| = 50/24.5 = 100/49</p><p>∴ Answer: D</p>
Correct Answer: D