If the mean deviation about the median of the numbers a, 2a, ....., 50a is 50, then |a| equals :
(1) 2
(2) 3
(3) 4
(4) 5

Options

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

Explanation

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 Numbers**:
The numbers are a, 2a, 3a, ..., 50a. This is an arithmetic sequence 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.
- 25th term = 25a
- 26th term = 26a
- Median = (25a + 26a) / 2 = (51a) / 2 = 25.5a

3. **Calculate the Mean Deviation**:
The mean deviation about the median is given by:
\[ M.D. = \frac{1}{n} \sum_{i=1}^{n} |x_i - \text{median}| \]
where \( n = 50 \) and \( x_i \) are the terms a, 2a, ..., 50a.

Thus, we need to calculate:
\[ M.D. = \frac{1}{50} \left( |a - 25.5a| + |2a - 25.5a| + |3a - 25.5a| + \ldots + |50a - 25.5a| \right) \]

4. **Simplify the Absolute Values**:
- For \( i = 1 \) to \( 25 \) (i.e., a to 25a):
\[ |ia - 25.5a| = |(i - 25.5)a| = (25.5 - i)a \]
- For \( i = 26 \) to \( 50 \) (i.e., 26a to 50a):
\[ |ia - 25.5a| = |(i - 25.5)a| = (i - 25.5)a \]

5. **Sum the Deviations**:
- For \( i = 1 \) to \( 25 \):
\[ \sum_{i=1}^{25} (25.5 - i)a = 25.5 \times 25a - \sum_{i=1}^{25} ia \]
The sum of the first 25 natural numbers is \( \frac{25 \times 26}{2} = 325 \).
Thus, the total for \( i = 1 \) to \( 25 \) is:
\[ 25.5 \times 25a - 325a = (637.5 - 325)a = 312.5a \]

- For \( i = 26 \) to \( 50 \):
\[ \sum_{i=26}^{50} (i - 25.5)a = \sum_{i=26}^{50} ia - 25.5 \times 25a \]
The sum of the first 50 natural numbers is \( \frac{50 \times 51}{2} = 1275 \).
Thus, the total for \( i = 26 \) to \( 50 \) is:
\[ 1275a - 637.5a = 637.5a \]

6. **Combine the Results**:
The total deviation is:
\[ 312.5a + 637.5a = 950a \]

7. **Calculate the Mean Deviation**:
\[ M.D. = \frac{950a}{50} = 19a \]

8. **Set the Mean Deviation Equal to 50**:
\[ 19a = 50 \implies a = \frac{50}{19} \]

9. **Find |a|**:
\[ |a| = \frac{50}{19} \approx 2.63 \]

Since the options provided are integers, we can check the closest integer value. The closest integer to \( 2.63 \) is 3.

### Final Answer:
Thus, the value of |a| is 3.