If the variance of first n even natural numbers is 133, then the value of n is equal to

Options

A. 19
B. 24
C. 21
D. 20

Answer

To find the value of \( n \) such that the variance of the first \( n \) even natural numbers is 133, we can follow these steps:

### Step 1: Identify the first \( n \) even natural numbers
The first \( n \) even natural numbers are:
\[ 2, 4, 6, \ldots, 2n \]

### Step 2: Calculate the mean of the first \( n \) even natural numbers
The mean \( \bar{x} \) of these numbers can be calculated as:
\[ \bar{x} = \frac{2 + 4 + 6 + \ldots + 2n}{n} = \frac{2(1 + 2 + 3 + \ldots + n)}{n} = \frac{2 \cdot \frac{n(n + 1)}{2}}{n} = n + 1 \]

### Step 3: Calculate the variance
The variance \( \sigma^2 \) is given by:
\[ \sigma^2 = \frac{1}{n} \sum_{i=1}^{n} (x_i - \bar{x})^2 \]
This can be rewritten using the formula:
\[ \sigma^2 = \frac{1}{n} \left( \sum_{i=1}^{n} x_i^2 - n \bar{x}^2 \right) \]

### Step 4: Calculate \( \sum_{i=1}^{n} x_i^2 \)
The sum of squares of the first \( n \) even natural numbers is:
\[ \sum_{i=1}^{n} (2i)^2 = 4 \sum_{i=1}^{n} i^2 = 4 \cdot \frac{n(n + 1)(2n + 1)}{6} = \frac{2n(n + 1)(2n + 1)}{3} \]

### Step 5: Substitute into the variance formula
Now substituting back into the variance formula:
\[ \sigma^2 = \frac{1}{n} \left( \frac{2n(n + 1)(2n + 1)}{3} - n(n + 1)^2 \right) \]
Simplifying this:
\[ \sigma^2 = \frac{1}{n} \left( \frac{2n(n + 1)(2n + 1) - 3n(n + 1)^2}{3} \right) \]
\[ = \frac{1}{3} \left( \frac{2(2n + 1) - 3(n + 1)}{n} \right) \]
\[ = \frac{1}{3} \left( \frac{4n + 2 - 3n - 3}{n} \right) = \frac{1}{3} \left( \frac{n - 1}{n} \right) \]

### Step 6: Set the variance equal to 133
Given that the variance is 133:
\[ \frac{1}{3} \cdot \frac{n - 1}{n} = 133 \]
Multiplying both sides by 3:
\[ \frac{n - 1}{n} = 399 \]
Cross multiplying gives:
\[ n - 1 = 399n \]
\[ n - 399n = 1 \]
\[ -398n = 1 \implies n = \frac{1}{398} \]
This does not make sense, so we need to re-evaluate our calculations.

### Step 7: Correct the variance calculation
We should have:
\[ \frac{1}{n} \left( \frac{2n(n + 1)(2n + 1)}{3} - n(n + 1)^2 \right) = 133 \]
This simplifies to:
\[ \frac{2n(n + 1)(2n + 1) - 3n(n + 1)^2}{3n} = 133 \]
Multiply through by \( 3n \):
\[ 2n(n + 1)(2n + 1) - 3n(n + 1)^2 = 399n \]
This leads to a polynomial in \( n \) which can be solved for \( n \).

### Step 8: Solve the polynomial
After solving the polynomial, we find that \( n = 20 \).

### Final Answer
Thus, the value of \( n \) is:
\[ \boxed{20} \]