If the mean deviation about the median of the numbers a, 2a ,…., 50 a is 50 then |a| equals

Options

A. 2
B. 3
C. 4
D. 5

Answer

To solve the problem, we need to find the value of |a| given that the mean deviation about the median of the numbers a, 2a, ..., 50a is 50.

### Step-by-Step Solution:

1. **Identify the Series**: The given series is a, 2a, 3a, ..., 50a. This is an arithmetic series with 50 terms.

2. **Find the Median**:
- Since there are 50 terms (even number), the median is the average of the 25th and 26th terms.
- The 25th term is 25a and the 26th term is 26a.
- Therefore, the median \( M \) is:
\[ M = \frac{25a + 26a}{2} = \frac{51a}{2} \]

3. **Calculate the Mean Deviation**:
- The mean deviation about the median is given by the formula:
\[ \text{Mean Deviation} = \frac{\sum |x_i - M|}{n} \]
- Here, \( n = 50 \) and \( M = \frac{51a}{2} \).
- We need to calculate \( |x_i - M| \) for each term \( x_i \) in the series.

4. **Compute \( |x_i - M| \)**:
- For \( x_i = ka \) where \( k = 1, 2, ..., 50 \):
\[ |x_i - M| = \left| ka - \frac{51a}{2} \right| = \left| a \left( k - \frac{51}{2} \right) \right| = |a| \left| k - 25.5 \right| \]

5. **Sum of Deviations**:
- We need to calculate \( \sum |x_i - M| \):
\[ \sum |x_i - M| = |a| \sum_{k=1}^{50} \left| k - 25.5 \right| \]
- The sum \( \sum_{k=1}^{50} |k - 25.5| \) can be split into two parts:
- For \( k = 1 \) to \( 25 \): \( |k - 25.5| = 25.5 - k \)
- For \( k = 26 \) to \( 50 \): \( |k - 25.5| = k - 25.5 \)

6. **Calculate Each Part**:
- For \( k = 1 \) to \( 25 \):
\[ \sum_{k=1}^{25} (25.5 - k) = 25.5 \times 25 - \sum_{k=1}^{25} k = 637.5 - \frac{25 \times 26}{2} = 637.5 - 325 = 312.5 \]
- For \( k = 26 \) to \( 50 \):
\[ \sum_{k=26}^{50} (k - 25.5) = \sum_{k=26}^{50} k - 25.5 \times 25 = \frac{50 \times 51}{2} - 25.5 \times 25 = 1275 - 637.5 = 637.5 \]

7. **Total Sum of Deviations**:
- Therefore, the total sum is:
\[ \sum |x_i - M| = |a| (312.5 + 637.5) = |a| \times 950 \]

8. **Mean Deviation Calculation**:
- Now, substituting back into the mean deviation formula:
\[ \text{Mean Deviation} = \frac{|a| \times 950}{50} = 19|a| \]
- We know from the problem statement that this mean deviation equals 50:
\[ 19|a| = 50 \]

9. **Solve for |a|**:
- Rearranging gives:
\[ |a| = \frac{50}{19} \]

### Final Answer:
\[ |a| = \frac{50}{19} \]