1. Key Concepts and Formulas

• Definition of Variance: For a set of $N$ observations $x_1, x_2, \ldots, x_N$, the variance (denoted by $V$ or $\sigma^2$) is a measure of how spread out the numbers are from their mean. It is calculated as the average of the squared differences from the mean: $V = \frac{\sum_{i=1}^{N} (x_i - \bar{x})^2}{N} = \frac{\sum x_i^2}{N} - (\bar{x})^2$ where $\bar{x}$ is the mean of the observations.
• Mean of First n Natural Numbers: For the sequence $1, 2, 3, \ldots, n$, the mean $\bar{x}$ is given by: $\bar{x} = \frac{n+1}{2}$
• Variance of First n Natural Numbers: A standard result in statistics for the variance of the first $n$ natural numbers ($1, 2, \ldots, n$) is: $V = \frac{n^2 - 1}{12}$ This formula is derived by substituting the sum of squares ($\sum x_i^2 = \frac{n(n+1)(2n+1)}{6}$) and the mean ($\bar{x} = \frac{n+1}{2}$) into the general variance formula.
• Problem Constraints: The problem specifies that $n$ is an odd natural number. This condition must be satisfied by our final answer.

2. Step-by-Step Solution

Step 1: Understand the Problem and Identify Given Information We are given a sequence of the first $n$ natural numbers: $1, 2, 3, \ldots, n$. We are also given that the variance of this sequence is $14$. Our goal is to find the value of $n$, ensuring it satisfies the condition of being an odd natural number.

Step 2: Apply the Variance Formula We will use the established formula for the variance of the first $n$ natural numbers. The given variance is $V = 14$. The formula for the variance of $1, 2, \ldots, n$ is: $V = \frac{n^2 - 1}{12}$

Step 3: Set up and Solve the Equation for n Substitute the given variance into the formula to form an equation: $\frac{n^2 - 1}{12} = 14$ Now, we solve this algebraic equation for $n$:

• Multiply both sides by 12 to clear the denominator: $n^2 - 1 = 14 \times 12$ $n^2 - 1 = 168$
• Add 1 to both sides to isolate the $n^2$ term: $n^2 = 168 + 1$ $n^2 = 169$
• Take the square root of both sides to find $n$: $n = \pm\sqrt{169}$ $n = \pm 13$

Step 4: Verify the Solution against Conditions The problem states that $n$ must be an odd natural number.

• A natural number must be a positive integer ($1, 2, 3, \ldots$). This means $n = -13$ is not a valid solution.
• We are left with $n = 13$.
• Now, we check if $n=13$ satisfies both conditions:
  • Is $13$ a natural number? Yes, it is a positive integer.
  • Is $13$ an odd number? Yes, it is not divisible by $2$. Since $n=13$ satisfies both conditions, it is the correct value.

3. Common Mistakes & Tips

• Formula Recall: Forgetting or incorrectly recalling the variance formula for the first $n$ natural numbers is a common pitfall. Memorizing standard formulas saves time and reduces calculation errors.
• Algebraic Errors: Mistakes in basic arithmetic operations (multiplication, addition, square roots) can lead to an incorrect answer. Double-check your calculations.
• Ignoring Conditions: Always pay close attention to the conditions specified for the variables in the problem (e.g., "odd", "natural number", "integer", "positive"). Failing to apply these conditions can lead to selecting an algebraically correct but contextually incorrect answer (like $n=-13$ in this case).

4. Summary

This problem required us to apply the known formula for the variance of the first $n$ natural numbers. By setting the formula equal to the given variance, we formed an algebraic equation. Solving this equation yielded $n=\pm 13$. Finally, we used the problem's constraint that $n$ must be an odd natural number to select the appropriate solution, $n=13$.

5. Final Answer

The final answer is $\boxed{13}$.