Here is the clear, educational, and well-structured solution:

1. Key Concepts and Formulas

   • Variance ($\sigma^2$): A measure of how spread out a set of data is from its mean. The computational formula is highly efficient: $\sigma^2 = \frac{\sum_{i=1}^{N} x_i^2}{N} - \left(\frac{\sum_{i=1}^{N} x_i}{N}\right)^2 = \overline{x^2} - (\bar{x})^2$ where $N$ is the number of observations, $\bar{x}$ is the mean of the observations, and $\overline{x^2}$ is the mean of the squares of the observations.
   • Sum of the first $n$ natural numbers: $\sum_{k=1}^{n} k = 1 + 2 + \dots + n = \frac{n(n+1)}{2}$
   • Sum of the squares of the first $n$ natural numbers: $\sum_{k=1}^{n} k^2 = 1^2 + 2^2 + \dots + n^2 = \frac{n(n+1)(2n+1)}{6}$
   • Variance of an Arithmetic Progression (AP): For an AP with $N$ terms and common difference $d$, the variance is given by the direct formula: $\text{Variance} = \frac{d^2(N^2-1)}{12}$.

2. Step-by-Step Solution

   Step 1: Identify the data set and number of observations. The problem asks for the variance of the first 50 even natural numbers.

   • What we are doing: Listing the data points and determining the total count.
   • Why we are doing this: To correctly identify the values $x_i$ and the number of observations $N$ required for the variance formula. The sequence is $2, 4, 6, \dots$. The $k$-th even natural number is $2k$. So, the first 50 even natural numbers are: $x_1 = 2 \times 1 = 2$ $x_2 = 2 \times 2 = 4$ ... $x_{50} = 2 \times 50 = 100$ Our data set is $\{2, 4, 6, \dots, 100\}$. The total number of observations, $N$, is $50$.

   Step 2: Calculate the sum of observations ($\sum x_i$).

   • What we are doing: Summing all the numbers in our data set.
   • Why we are doing this: This sum is needed to calculate the mean ($\bar{x}$) of the data. We need to find $\sum x_i = 2 + 4 + 6 + \dots + 100$. Factor out 2 from each term to simplify into a known series: $\sum x_i = 2(1 + 2 + 3 + \dots + 50)$ Using the formula for the sum of the first $n$ natural numbers with $n=50$: $\sum x_i = 2 \times \frac{50(50+1)}{2}$ $\sum x_i = 2 \times \frac{50 \times 51}{2}$ $\sum x_i = 50 \times 51 = 2550$

   Step 3: Calculate the mean of observations ($\bar{x}$) and its square.

   • What we are doing: Finding the average of the data points and then squaring it.
   • Why we are doing this: The variance formula requires the term $(\bar{x})^2$. Using the sum from Step 2: $\bar{x} = \frac{\sum x_i}{N} = \frac{2550}{50}$ $\bar{x} = 51$ Now, square the mean: $(\bar{x})^2 = (51)^2 = 2601$

   Step 4: Calculate the sum of squares of observations ($\sum x_i^2$).

   • What we are doing: Squaring each number in the data set and then summing these squares.
   • Why we are doing this: This sum is needed to calculate the mean of squares ($\overline{x^2}$), which is a component of the variance formula. We need to find $\sum x_i^2 = 2^2 + 4^2 + 6^2 + \dots + 100^2$. Factor out $2^2=4$ from each term to simplify into a known series: $\sum x_i^2 = 2^2(1^2 + 2^2 + 3^2 + \dots + 50^2)$ $\sum x_i^2 = 4(1^2 + 2^2 + 3^2 + \dots + 50^2)$ Using the formula for the sum of the squares of the first $n$ natural numbers with $n=50$: $\sum x_i^2 = 4 \times \frac{50(50+1)(2 \times 50+1)}{6}$ $\sum x_i^2 = 4 \times \frac{50 \times 51 \times 101}{6}$ Simplify the expression: $\sum x_i^2 = \frac{2 \times 50 \times 51 \times 101}{3}$ $\sum x_i^2 = 100 \times \frac{51}{3} \times 101$ $\sum x_i^2 = 100 \times 17 \times 101$ $\sum x_i^2 = 1700 \times 101 = 171700$

   Step 5: Calculate the mean of squares of observations ($\overline{x^2}$).

   • What we are doing: Finding the average of the squared data points.
   • Why we are doing this: This is the $\overline{x^2}$ term required for the variance formula. Using the sum of squares from Step 4: $\overline{x^2} = \frac{\sum x_i^2}{N} = \frac{171700}{50}$ $\overline{x^2} = \frac{17170}{5} = 3434$

   Step 6: Apply the variance formula.

   • What we are doing: Substituting the calculated mean of squares and the square of the mean into the variance formula.
   • Why we are doing this: To obtain the final variance value. Using the values from Step 3 and Step 5: $\text{Variance} (\sigma^2) = \overline{x^2} - (\bar{x})^2$ $\text{Variance} (\sigma^2) = 3434 - 2601$ $\text{Variance} (\sigma^2) = 833$

   Step 7: Verify using the Variance of an Arithmetic Progression (AP) formula.

   • What we are doing: Applying a specialized shortcut formula for the variance of an arithmetic progression.
   • Why we are doing this: To quickly confirm the result obtained from the longer, fundamental method, which is a good practice in competitive exams. Our data set $\{2, 4, 6, \dots, 100\}$ is an AP with:
   • Number of terms, $N = 50$
   • Common difference, $d = 2$ Using the formula $\text{Variance} = \frac{d^2(N^2-1)}{12}$: $\text{Variance} = \frac{2^2(50^2-1)}{12}$ $\text{Variance} = \frac{4(2500-1)}{12}$ $\text{Variance} = \frac{4 \times 2499}{12}$ $\text{Variance} = \frac{2499}{3}$ $\text{Variance} = 833$ Both methods yield the same result, confirming our calculation.

3. Common Mistakes & Tips

   • Arithmetic Errors: Be extremely careful with calculations, especially when dealing with large numbers. Double-check multiplications and divisions.
   • Incorrect Formulas: Ensure you use the correct formulas for the sum of natural numbers and sum of squares. A common mistake is using the sum of $n$ for sum of $n^2$.
   • Misinterpreting the Data Set: Read the question carefully. "First 50 even natural numbers" is different from "first 50 natural numbers" or "first 50 odd natural numbers."
   • Leverage AP Formula: For data sets that form an Arithmetic Progression, remember and use the direct variance formula ($\frac{d^2(N^2-1)}{12}$). It's a powerful time-saver and a great way to cross-verify answers.

4. Summary

   To find the variance of the first 50 even natural numbers, we first identified the data set and the number of observations ($N=50$). We then systematically calculated the sum of observations ($\sum x_i$) and the sum of squares of observations ($\sum x_i^2$) using standard summation formulas. From these, we derived the mean ($\bar{x}$) and the mean of squares ($\overline{x^2}$). Finally, we applied the variance formula $\sigma^2 = \overline{x^2} - (\bar{x})^2$ to get the result. An alternative method using the specific variance formula for an arithmetic progression confirmed the answer, yielding 833.

The final answer is $\boxed{\text{833}}$, which corresponds to option (A).