<p>If the mean deviation of the numbers 1, 1 + <em>d</em>, 1 + 2<em>d</em>, ..., 1 + 100<em>d</em> from their mean is 225, then <em>d</em> is equal to</p>
Step-by-Step Solution
Key Concept: The mean deviation depends only on |d|, not the sign of d. For an arithmetic progression with 101 terms, the mean equals the middle term (1 + 50d), and deviations form a symmetric pattern allowing us to sum only half the terms.
<p><strong>Step 1:</strong> Identify the AP: 1, 1+d, 1+2d, ..., 1+100d has 101 terms.</p><p><strong>Step 2:</strong> Find the mean: Mean = (sum of all terms)/101 = [101 + d(0+1+2+...+100)]/101 = [101 + d·5050]/101 = 1 + 50d</p><p><strong>Step 3:</strong> Calculate deviations from mean. The k-th term is 1+kd (k=0,1,...,100), so deviation = |1+kd - (1+50d)| = |kd - 50d| = |d|·|k-50|</p><p><strong>Step 4:</strong> Sum all absolute deviations:</p><p>MD = (1/101)·Σ|d|·|k-50| for k=0 to 100</p><p>= (|d|/101)·[|0-50| + |1-50| + |2-50| + ... + |100-50|]</p><p>= (|d|/101)·[50 + 49 + 48 + ... + 1 + 0 + 1 + 2 + ... + 50]</p><p>= (|d|/101)·2(1+2+...+50)</p><p>= (|d|/101)·2·[50·51/2]</p><p>= (|d|/101)·2550</p><p><strong>Step 5:</strong> Set MD = 225:</p><p>(|d|/101)·2550 = 225</p><p>|d| = (225·101)/2550 = 22725/2550 = 9</p><p>∴ d = ±9 (Answer: C)</p>
Correct Answer: C