<p>If the mean deviation about the median of the numbers \(a, 2a, \ldots, 50a\) is 50, then \(|a|\) equals</p>
Step-by-Step Solution
Key Concept: Mean deviation about median = (sum of absolute deviations from median) / n. For the sequence a, 2a, ..., 50a with 50 terms, the median is (25a + 26a)/2 = 25.5a, and by symmetry, the sum of deviations simplifies to a predictable form based on |a|.
<p><strong>Step 1:</strong> Identify the sequence: a, 2a, 3a, ..., 50a (50 terms)</p><p><strong>Step 2:</strong> Find the median. Since there are 50 terms (even), the median is M = (25th term + 26th term)/2 = (25a + 26a)/2 = 25.5a</p><p><strong>Step 3:</strong> Calculate mean deviation. For each term ka, the deviation from median is |ka - 25.5a| = |a||k - 25.5|</p><p><strong>Step 4:</strong> Use symmetry. The deviations are |a|(|-24.5|, |-23.5|, ..., |-0.5|, |0.5|, ..., |23.5|, |24.5|). By symmetry: MD = |a| × [2(0.5 + 1.5 + 2.5 + ... + 24.5)]/50</p><p><strong>Step 5:</strong> Evaluate the sum: 0.5 + 1.5 + 2.5 + ... + 24.5 = 0.5(1 + 3 + 5 + ... + 49) = 0.5 × 25² = 312.5</p><p><strong>Step 6:</strong> MD = |a| × (2 × 312.5)/50 = |a| × 625/50 = 12.5|a|</p><p><strong>Step 7:</strong> Given MD = 50: 12.5|a| = 50 → |a| = 4</p><p>∴ Answer: B (|a| = 4)</p>
Correct Answer: B