1. Key Concepts and Formulas

• Median (M): For a dataset arranged in ascending or descending order, the median is the middle value.
  • If the number of terms ($N$) is odd, $M = \left(\frac{N+1}{2}\right)$-th term.
  • If $N$ is even, $M = \frac{\left(\frac{N}{2}\right)\text{-th term} + \left(\frac{N}{2}+1\right)\text{-th term}}{2}$.
• Mean Deviation about Median (M.D.): This measures the average absolute difference of data points from the median. $\text{M.D.} = \frac{\sum_{i=1}^{N} |x_i - M|}{N}$ where $x_i$ are the data points, $M$ is the median, and $N$ is the total number of terms.
• Sum of an Arithmetic Progression (A.P.): For an A.P. with $k$ terms, first term $A$, and last term $L$, the sum $S_k = \frac{k}{2}(A+L)$.

2. Step-by-Step Solution

Step 1: Analyze the Given Data Set and Determine N The given numbers are $a, 2a, 3a, \ldots, 50a$.

• We can clearly see there are 50 terms in this sequence.
• Therefore, the total number of data points, $N = 50$.
• For calculating the median, the data must be ordered. If $a > 0$, the terms are already in ascending order. If $a < 0$, the order would be reversed ($50a, 49a, \ldots, a$). However, the positions of the median terms remain the same, and the absolute value in the mean deviation formula will correctly handle the sign of $a$.

Step 2: Calculate the Median (M) Since $N=50$ is an even number, the median is the average of the $\left(\frac{N}{2}\right)$-th term and the $\left(\frac{N}{2}+1\right)$-th term.

• The $\frac{N}{2}$-th term is the $\frac{50}{2} = 25$-th term, which is $x_{25} = 25a$.
• The $\left(\frac{N}{2}+1\right)$-th term is the $\left(\frac{50}{2}+1\right) = 26$-th term, which is $x_{26} = 26a$. Now, we calculate the median $M$: $M = \frac{x_{25} + x_{26}}{2}$ $M = \frac{25a + 26a}{2}$ $M = \frac{51a}{2}$ $M = 25.5a$ So, the median of the dataset is $25.5a$.

Step 3: Calculate the Sum of Absolute Deviations from the Median Next, we need to find $\sum_{i=1}^{N} |x_i - M|$. Substituting $x_i = ia$ and $M = 25.5a$: $\sum_{i=1}^{50} |ia - 25.5a|$ We can factor out $a$ from inside the absolute value. Using the property $|xy| = |x||y|$: $\sum_{i=1}^{50} |(i - 25.5)a| = |a| \sum_{i=1}^{50} |i - 25.5|$ Now, let's evaluate the sum $\sum_{i=1}^{50} |i - 25.5|$. The term $(i - 25.5)$ changes sign at $i=25.5$.

• For $i = 1, 2, \ldots, 25$: $(i - 25.5)$ is negative, so $|i - 25.5| = -(i - 25.5) = 25.5 - i$.
• For $i = 26, 27, \ldots, 50$: $(i - 25.5)$ is positive, so $|i - 25.5| = i - 25.5$.

We can split the sum into two parts: $\sum_{i=1}^{25} (25.5 - i) + \sum_{i=26}^{50} (i - 25.5)$ Let's list the terms to observe the pattern: First part: $(25.5-1) + (25.5-2) + \ldots + (25.5-25) = 24.5 + 23.5 + \ldots + 0.5$ Second part: $(26-25.5) + (27-25.5) + \ldots + (50-25.5) = 0.5 + 1.5 + \ldots + 24.5$ Notice the symmetry: the two sums are identical. So, the total sum is: $2 \times (0.5 + 1.5 + 2.5 + \ldots + 24.5)$ This is an Arithmetic Progression (A.P.) with:

• First term ($A$) $= 0.5$
• Last term ($L$) $= 24.5$
• Common difference ($d$) $= 1.0$ To find the number of terms ($k$) in this A.P., we use $L = A + (k-1)d$: $24.5 = 0.5 + (k-1)1$ $24 = k-1 \implies k = 25$ Now, calculate the sum of this A.P. using $S_k = \frac{k}{2}(A+L)$: $S_{25} = \frac{25}{2}(0.5 + 24.5) = \frac{25}{2}(25) = \frac{625}{2}$ Substitute this back into the expression for the total sum of absolute deviations: $\sum_{i=1}^{50} |x_i - M| = |a| \times 2 \times S_{25} = |a| \times 2 \times \frac{625}{2} = 625|a|$

Step 4: Apply the Mean Deviation Formula and Solve for $|a|$ We are given that the mean deviation about the median is 50. Using the formula: $\text{M.D.} = \frac{\sum_{i=1}^{N} |x_i - M|}{N}$ Substitute M.D. = 50, $N = 50$, and $\sum_{i=1}^{N} |x_i - M| = 625|a|$: $50 = \frac{625|a|}{50}$ Now, solve for $|a|$: $50 \times 50 = 625|a|$ $2500 = 625|a|$ $|a| = \frac{2500}{625}$ $|a| = 4$

3. Common Mistakes & Tips

• Median Calculation: Be careful when $N$ is even; it's the average of the two middle terms, not just one.
• Absolute Value Property: Remember that $|xy| = |x||y|$. Factoring out $|a|$ is crucial, especially since $a$ could be negative.
• Splitting the Sum: Correctly identify the point where the term inside the absolute value changes sign to properly remove the absolute value bars.
• Arithmetic Progression: Recognize and correctly apply the formula for the sum of an arithmetic progression to simplify calculations.

4. Summary

To find $|a|$, we first identified the number of terms ($N=50$) and calculated the median ($M=25.5a$) since $N$ is even. Then, we computed the sum of absolute deviations from the median, $\sum |x_i - M|$, by factoring out $|a|$ and splitting the sum into two symmetric arithmetic progressions. This sum evaluated to $625|a|$. Finally, we used the given mean deviation of 50 in the formula $\text{M.D.} = \frac{\sum |x_i - M|}{N}$ to solve for $|a|$, yielding 4.

5. Final Answer

The final answer is $\boxed{4}$, which corresponds to option (A).