Question

Mean deviation about median for \( k, 2k, 3k, \ldots, 1000k \) is 500, then the value of \( k^2 \) is:

• A. \(4\)
• B. \(9\)
• C. \(16\)
• D. \(1\)

Hint:

For symmetric data:
Median lies at the center
Mean deviation can be calculated efficiently using symmetry

Answer 1

The Correct Option is A

Concept: Mean deviation about median is defined as: \[ \text{Mean Deviation} = \frac{1}{n}\sum |x_i - \text{Median}| \] For an arithmetic sequence with an even number of terms, the median is the average of the two middle terms.
Step 1: Identify the median. The given sequence is: \[ k, 2k, 3k, \ldots, 1000k \] Number of terms \(n = 1000\) (even). Median: \[ \text{Median} = \frac{500k + 501k}{2} = \frac{1001k}{2} \]
Step 2: Write the general term and deviation from median. General term: \[ x_i = ik \] Deviation from median: \[ |x_i - \text{Median}| = \left| ik - \frac{1001k}{2} \right| = k\left| i - \frac{1001}{2} \right| \]
Step 3: Use symmetry to find total deviation. Due to symmetry about the median: \[ \sum |x_i - \text{Median}| = 2k \sum_{i=1}^{500} \left(\frac{1001}{2} - i\right) \] The terms form an A.P. from \(499.5\) to \(0.5\). Number of terms \(= 500\) Average term: \[ \frac{499.5 + 0.5}{2} = 250 \] Sum: \[ 500 \times 250 = 125000 \] Thus, \[ \sum |x_i - \text{Median}| = 2k \times 125000 = 250000k \]
Step 4: Calculate mean deviation. \[ \text{Mean Deviation} = \frac{250000k}{1000} = 250k \] Given: \[ 250k = 500 \Rightarrow k = 2 \]
Step 5: Find \(k^2\). \[ k^2 = 4 \] \[ \boxed{k^2 = 4} \]