Question

The variance of first $50$ even natural numbers is

• A. $ \frac{ 833}{ 4}$
• B. $833$
• C. $437$
• D. $ \frac{437}{4}$

Answer 1

The Correct Option is B

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

1. List the first 50 even natural numbers:

   The first 50 even natural numbers are: 2, 4, 6, ..., 100. These numbers form an arithmetic sequence with the first term a = 2 and the common difference d = 2.

2. Find the sum of the first 50 even numbers:

   The sum of an arithmetic series is given by:

   S_n = \frac{n}{2} \cdot (2a + (n-1)d)

   Substitute the values:

   S_{50} = \frac{50}{2} \cdot (2 \times 2 + (50-1) \times 2)

   = 25 \cdot (4 + 98)

   = 25 \cdot 102

   = 2550

3. Compute the mean of the first 50 even numbers:

   \text{Mean} = \frac{S_{50}}{50} = \frac{2550}{50} = 51

4. Calculate the sum of the squares of the first 50 even numbers:

   Each term in the series can be represented as a_i = 2i for i = 1 \text{ to } 50. Therefore, the sum of squares is:

   \sum_{i=1}^{50} (2i)^2 = 4 \sum_{i=1}^{50} i^2

   The formula for the sum of squares of the first n natural numbers is:

   \sum_{i=1}^{n} i^2 = \frac{n(n+1)(2n+1)}{6}

   Substitute n = 50:

   \sum_{i=1}^{50} i^2 = \frac{50 \cdot 51 \cdot 101}{6}

   = 42925

   Then, \sum_{i=1}^{50} (2i)^2 = 4 \cdot 42925 = 171700

5. Calculate the variance:

   The formula for variance is:

   \sigma^2 = \frac{1}{n} \cdot \sum (x_i^2) - \text{Mean}^2

   Substitute the known values:

   \sigma^2 = \frac{1}{50} \cdot 171700 - 51^2

   = 3434 - 2601

   = 833

Therefore, the variance of the first 50 even natural numbers is 833.