The variance of first 20- natural numbers is

Options

A. `133//4`
B. ` 379//12`
C. ` 133//2`
D. ` 399//4`

Answer

To find the variance of the first 20 natural numbers, we can use the formula for the variance of the first N natural numbers, which is given by:

\[ \text{Variance} = \frac{N^2 - 1}{12} \]

where \( N \) is the number of natural numbers.

### Step-by-Step Solution:

1. **Identify N**:
We are asked to find the variance for the first 20 natural numbers, so we have:
\[ N = 20 \]

2. **Calculate \( N^2 \)**:
We need to calculate \( N^2 \):
\[ N^2 = 20^2 = 400 \]

3. **Substitute into the Variance Formula**:
Now, substitute \( N^2 \) into the variance formula:
\[ \text{Variance} = \frac{400 - 1}{12} \]

4. **Perform the Subtraction**:
Calculate \( 400 - 1 \):
\[ 400 - 1 = 399 \]

5. **Divide by 12**:
Now, divide by 12:
\[ \text{Variance} = \frac{399}{12} \]

6. **Simplify the Fraction**:
To simplify \( \frac{399}{12} \), we can check if it can be reduced. The greatest common divisor of 399 and 12 is 3:
\[ \frac{399 \div 3}{12 \div 3} = \frac{133}{4} \]

Thus, the variance of the first 20 natural numbers is:
\[ \text{Variance} = \frac{133}{4} \]

### Final Answer:
The correct option is \( \frac{133}{4} \).

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