1. Key Concepts and Formulas

To solve this problem, we need to understand the definitions and formulas for the mean and mean deviation about the mean for a dataset.

• Mean ($\bar{x}$): The average of a set of $n$ observations $x_1, x_2, \dots, x_n$ is given by: $\bar{x} = \frac{\sum_{i=1}^{n} x_i}{n}$ For the first $n$ natural numbers ($1, 2, \dots, n$), the sum is $\sum_{i=1}^{n} i = \frac{n(n+1)}{2}$.
• Mean Deviation about the Mean (M.D.): This measures the average absolute difference between each observation and the mean. It is calculated as: $\text{M.D.} = \frac{\sum_{i=1}^{n} |x_i - \bar{x}|}{n}$ The absolute value ensures that both positive and negative deviations contribute positively to the total spread.

2. Step-by-Step Solution

Let's break down the problem into logical steps, carefully calculating each component.

Step 1: Calculate the Mean ($\bar{x}$)

• What and Why: The mean is the central point of our data, and it's the first value needed to calculate deviations. Our dataset is $x_i = 1, 2, \dots, n$.
• Math: The sum of the first $n$ natural numbers is $\sum_{i=1}^{n} x_i = \frac{n(n+1)}{2}$. Now, apply the mean formula: $\bar{x} = \frac{\sum_{i=1}^{n} x_i}{n} = \frac{\frac{n(n+1)}{2}}{n}$ $\bar{x} = \frac{n+1}{2}$
• Reasoning: Since $n$ is given as an odd integer, $n+1$ will always be an even integer. Therefore, $\frac{n+1}{2}$ will be an integer, which is the exact middle term of the sequence $1, 2, \dots, n$.

Step 2: Calculate the Absolute Deviations ($|x_i - \bar{x}|$ )

• What and Why: We need to find the absolute difference between each number in the dataset and the mean. This is crucial for computing the sum of absolute deviations.
• Math: We need to find $|x_i - \frac{n+1}{2}|$ for each $x_i$ from $1$ to $n$. The dataset is symmetric around the mean $\bar{x} = \frac{n+1}{2}$.
  • For terms $x_i < \bar{x}$: The deviations are $\bar{x} - x_i$. The terms are $1, 2, \dots, \frac{n-1}{2}$. The corresponding deviations are: $|\frac{n+1}{2} - 1| = \frac{n-1}{2}$ $|\frac{n+1}{2} - 2| = \frac{n-3}{2}$ ... $|\frac{n+1}{2} - \frac{n-1}{2}| = 1$ So, these deviations are $\frac{n-1}{2}, \frac{n-3}{2}, \dots, 1$.
  • For the term $x_i = \bar{x}$: The deviation is $0$. $|\frac{n+1}{2} - \frac{n+1}{2}| = 0$.
  • For terms $x_i > \bar{x}$: The deviations are $x_i - \bar{x}$. The terms are $\frac{n+1}{2}+1, \dots, n$. The corresponding deviations are: $|(\frac{n+1}{2}+1) - \frac{n+1}{2}| = 1$ ... $|n - \frac{n+1}{2}| = \frac{2n - (n+1)}{2} = \frac{n-1}{2}$ So, these deviations are $1, 2, \dots, \frac{n-1}{2}$.
• Reasoning: The full list of absolute deviations, arranged, is: $\left\{ \frac{n-1}{2}, \frac{n-3}{2}, \dots, 1, 0, 1, \dots, \frac{n-3}{2}, \frac{n-1}{2} \right\}$ This symmetrical pattern simplifies the sum significantly.

Step 3: Calculate the Sum of Absolute Deviations ($\sum |x_i - \bar{x}|$ )

• What and Why: This sum forms the numerator of the Mean Deviation formula. The symmetry we observed is key to an efficient calculation.
• Math: The sum consists of $0$ (from the mean itself) and two identical sets of deviations: $1, 2, \dots, \frac{n-1}{2}$. Let $k = \frac{n-1}{2}$. The sum of the first $k$ natural numbers is $\frac{k(k+1)}{2}$. $\sum_{i=1}^{n} |x_i - \bar{x}| = 2 \times \left( 1 + 2 + \dots + \frac{n-1}{2} \right) + 0$ Substitute $k = \frac{n-1}{2}$: $\sum_{i=1}^{n} |x_i - \bar{x}| = 2 \times \frac{\left(\frac{n-1}{2}\right) \left(\frac{n-1}{2} + 1\right)}{2}$ $= \left(\frac{n-1}{2}\right) \left(\frac{n-1+2}{2}\right)$ $= \left(\frac{n-1}{2}\right) \left(\frac{n+1}{2}\right)$ $\sum_{i=1}^{n} |x_i - \bar{x}| = \frac{(n-1)(n+1)}{4}$
• Reasoning: By recognizing the arithmetic series and its repetition due to symmetry, we avoid summing each individual deviation.

Step 4: Calculate the Mean Deviation about the Mean (M.D.)

• What and Why: Now that we have the sum of absolute deviations, we can apply the M.D. formula to find the average deviation.
• Math: Using the formula $\text{M.D.} = \frac{\sum_{i=1}^{n} |x_i - \bar{x}|}{n}$: $\text{M.D.} = \frac{\frac{(n-1)(n+1)}{4}}{n}$ $\text{M.D.} = \frac{(n-1)(n+1)}{4n}$
• Reasoning: This is a direct application of the M.D. definition.

Step 5: Solve for n

• What and Why: The problem provides an expression for the mean deviation. By equating our derived formula with the given expression, we can form an equation and solve for $n$.
• Math: We are given that the mean deviation is $\frac{5(n+1)}{n}$. Equate our calculated M.D. with the given M.D.: $\frac{(n-1)(n+1)}{4n} = \frac{5(n+1)}{n}$ To solve for $n$, we can multiply both sides by $4n$. Since $n$ is a natural number ($n \ge 1$), $n \ne 0$. Also, since $n \ge 1$, $n+1 \ne 0$. So, we can safely divide both sides by $(n+1)$ as well. $(n-1) = 5 \times 4$ $n-1 = 20$ $n = 21$
• Reasoning: The algebraic manipulation yields $n=21$. This value satisfies the condition that $n$ is an odd integer.

3. Common Mistakes & Tips

• Tip for Mean: For an arithmetic progression with an odd number of terms, the mean is simply the median (the middle term). This can save time and provide a good intuition.
• Common Mistake: Errors in calculating the sum of the series $1+2+\dots+k$. Ensure the upper limit $k$ is correctly identified as $\frac{n-1}{2}$ for the sum of deviations on one side of the mean.
• Tip for Symmetry: Always look for symmetry in statistical problems. It frequently simplifies sums of deviations, variances, and other calculations by allowing you to sum a smaller series and multiply by a factor.
• Algebraic Precision: Be careful when simplifying equations. Ensure that any terms you cancel from both sides are non-zero to avoid losing valid solutions.

4. Summary

This problem demonstrated the step-by-step process of calculating the mean deviation about the mean for an arithmetic progression of natural numbers. We first determined the mean, then leveraged the dataset's symmetry to efficiently compute the sum of absolute deviations. Finally, by equating our derived mean deviation formula with the given expression, we solved for $n$, confirming that the result $n=21$ satisfied all problem conditions.

5. Final Answer The final answer is $\boxed{21}$.