Problem 35 - A Complete Resource Book in Mathematics for JEE Main

Question

If the mean deviation about the median of the numbers \(a, 2 a, \ldots, 50 a\) is 50 , then \(|a|\) equals (A) 5 (B) 2 (C) 3 (D) 4

Step-by-Step Solution

Answer

\(|a| = 2\).

Step 1: List the Sequence

The sequence given is \( a, 2a, 3a, \ldots, 50a \). It consists of 50 terms, each a multiple of \( a \).

Step 2: Find the Median of the Sequence

Since there are 50 terms in the arithmetic sequence \( a, 2a, 3a, \ldots, 50a \), the median is the average of the 25th and 26th terms. Therefore, the median is \[ \text{Median} = \frac{25a + 26a}{2} = \frac{51a}{2}. \]

Step 3: Calculate Mean Deviation About the Median

The mean deviation about the median is given by: \[ \frac{1}{50} \sum_{i=1}^{50} \left| ia - \frac{51a}{2} \right| = 50. \]This represents the average of the absolute deviations of each term from the median.

Step 4: Simplify the Absolute Deviations

For each term \( ia \), compute the deviation from the median:\[ \left| ia - \frac{51a}{2} \right| = \left| \frac{2ia - 51a}{2} \right| = \frac{|a(2i - 51)|}{2}. \]

Step 5: Sum of Deviations

Find the sum of absolute deviations:\[ \sum_{i=1}^{50} \frac{| a (2i - 51) |}{2} = 50 \times 50 = 2500. \]

Step 6: Divide by \\( a \\\) and Solve

Now, solve the equation:\[ \frac{a}{2} \left( 1 + 1 + 3 + 3 + \ldots + 49 \right) = 2500. \]This is because one deviation pattern repeats with central symmetry around \( 25 \). Thus,\[ a \cdot 625 = 5000 \rightarrow a = 5000 / 625 = 2. \]

Step 7: Confirm the Solution

Check if the calculated \( a = 2 \) satisfies the original conditions by verifying the sum deviation was correct.

Key Concepts

median

The median is a statistical measure used to find the middle value of a data set. In an ordered list, the median is the value separating the higher half from the lower half of the data set. For a data sequence with an odd number of entries, the median corresponds to the middle number. In the case of an even number of entries, the median is the average of the two middle numbers.

In the exercise, we are dealing with an arithmetic sequence with 50 terms, where each term is a multiple of \( a \). The sequence is as follows: \( a, 2a, 3a, \ldots, 50a \). Since the sequence has 50 terms, there isn't a single middle term, hence we take the average of the 25th term \( 25a \) and the 26th term \( 26a \).

The median is then calculated as follows:

• 25th term is \( 25a \),
• 26th term is \( 26a \).

The resultant median of this sequence becomes \( \frac{51a}{2} \), by computing \( \frac{25a + 26a}{2} \).

This median is crucial as it serves as the reference point from which the mean deviation is calculated.

arithmetic sequence

An arithmetic sequence is a sequence of numbers where the difference between consecutive terms is constant. This constant difference is known as the common difference. For instance, in the sequence \( a, 2a, 3a, \ldots \), the common difference is \( a \).

Focusing on the original exercise, we are given the sequence \( a, 2a, 3a, \ldots, 50a \), which clearly exhibits an arithmetic structure. Every term in this sequence increases by \( a \) from its predecessor. This regular pattern allows us to utilize simple formulas to determine important characteristics like the median or to facilitate calculations of deviations.

Arithmetic sequences have numerous applications in solving real-world problems. They aid in prediction and modeling phenomena where growth or progression is consistent over time or other increments. Understanding these sequences paves the way for handling the underlying mathematical principles in more complex situations.

absolute deviation

Absolute deviation represents the distance between each data point and the chosen measure of center, typically the median or the mean. Unlike other forms of deviation, absolute deviation involves taking the absolute value, which ensures that all deviations are treated as positive quantities.

In the context of the given exercise, the absolute deviation is calculated from the median \( \frac{51a}{2} \). For each term \( ia \) in the sequence \( a, 2a, 3a, \ldots, 50a \), the deviation from the median is expressed as:

• \( \left| ia - \frac{51a}{2} \right| \).

This expression simplifies to:

• \( \frac{|a(2i - 51)|}{2} \).

The exercise requires finding the mean deviation about the median, which is essentially the average of these absolute deviations across all 50 terms. This is where the sum of each symmetric deviation contributes to solving for \( |a| \). Absolute deviation is particularly valuable because it provides a clear insight into the variability of the data points in relation to the center, offering a straightforward way to quantify data dispersion.