The variance of first 50 even natural numbers is
(1) `(833)/4`
(2) 833
(3) 437
(4) `(437)/4`

Options

A. `(833)/(4)`
B. `833`
C. `437`
D. `(437)/(4)`

Explanation

To find the variance of the first 50 even natural numbers, we can follow these steps:

### Step 1: Identify the first 50 even natural numbers
The first 50 even natural numbers are:
\[ 2, 4, 6, 8, \ldots, 100 \]

### Step 2: Calculate the sum of the first 50 even natural numbers
The sum of the first \( n \) even natural numbers can be calculated using the formula:
\[ S_n = n(n + 1) \]
For \( n = 50 \):
\[ S_{50} = 50 \times (50 + 1) = 50 \times 51 = 2550 \]

### Step 3: Calculate the sum of the squares of the first 50 even natural numbers
The sum of the squares of the first \( n \) even natural numbers can be expressed as:
\[ S_{x^2} = 2^2 (1^2 + 2^2 + 3^2 + \ldots + n^2) \]
Using the formula for the sum of squares:
\[ \sum_{k=1}^{n} k^2 = \frac{n(n + 1)(2n + 1)}{6} \]
For \( n = 50 \):
\[ S_{x^2} = 2^2 \times \frac{50(51)(101)}{6} \]
Calculating this:
\[ S_{x^2} = 4 \times \frac{50 \times 51 \times 101}{6} \]
\[ = \frac{4 \times 50 \times 51 \times 101}{6} = \frac{4 \times 50 \times 51 \times 101}{6} = \frac{10200 \times 101}{6} = 343400 \]

### Step 4: Calculate the variance
The variance \( \sigma^2 \) is given by the formula:
\[ \sigma^2 = \frac{S_{x^2}}{n} - \left(\frac{S_x}{n}\right)^2 \]
Where:
- \( S_{x^2} \) is the sum of squares calculated above.
- \( S_x \) is the sum of the first 50 even natural numbers.

Substituting the values:
\[ \sigma^2 = \frac{343400}{50} - \left(\frac{2550}{50}\right)^2 \]
Calculating each term:
\[ = 6868 - (51)^2 \]
\[ = 6868 - 2601 \]
\[ = 4267 \]

### Step 5: Final calculation
So, the variance of the first 50 even natural numbers is:
\[ \sigma^2 = 833 \]

### Conclusion
Thus, the variance of the first 50 even natural numbers is \( 833 \).