Solution:

Mean Deviation about Median:
The mean deviation about the median is a measure of the spread or dispersion of a set of numbers. It is calculated by taking the absolute difference between each number and the median, and then finding the average of these differences.

Given:
The numbers in the set are a, 2a, ..., 50a.

To find the mean deviation about the median, we first need to find the median.

Finding the Median:
The median is the middle value of a set of numbers when arranged in ascending order. Since the numbers in the set are in arithmetic progression with a common difference of a, the median can be found by taking the average of the middle two terms.

The middle two terms are the ((n + 1)/2)th term and ((n + 3)/2)th term, where n is the total number of terms.

In this case, n = 50, so the middle terms are the 25th term and the 26th term.

The 25th term is (25a), and the 26th term is (26a).

Therefore, the median = (25a + 26a)/2 = (51a)/2 = 25.5a.

Calculating the Mean Deviation about the Median:
To calculate the mean deviation about the median, we need to find the absolute difference between each number and the median, and then find the average of these differences.

The numbers in the set can be written as:
a, 2a, 3a, ..., 50a.

The absolute differences between each number and the median (25.5a) are:
|a - 25.5a|, |2a - 25.5a|, |3a - 25.5a|, ..., |50a - 25.5a|.

Simplifying these absolute differences, we get:
|24.5a|, |23.5a|, |22.5a|, ..., |24.5a|.

The sum of these absolute differences is:
|24.5a| + |23.5a| + |22.5a| + ... + |24.5a|.

Since there are 50 terms, the average absolute difference is:
(|24.5a| + |23.5a| + |22.5a| + ... + |24.5a|)/50.

Given that the mean deviation about the median is 50, we can set up the equation:
(|24.5a| + |23.5a| + |22.5a| + ... + |24.5a|)/50 = 50.

Simplifying the equation, we get:
(|24.5a| + |23.5a| + |22.5a| + ... + |24.5a|) = 2500.

Since all the terms in the absolute value signs are positive, we can remove the absolute value signs.

The equation becomes:
(24.5a + 23.5a + 22.5a + ... + 24.5a) = 2500.

Simplifying further, we get:
(24.5 + 23.5 + 22.5 + ... + 24.5)a = 2500.

The sum of an arithmetic series can be calculated using