Variance of 1st n natural number `1,2,3, . . n` is 14 then n is

Answer

To find the value of \( n \) such that the variance of the first \( n \) natural numbers \( 1, 2, 3, \ldots, n \) is equal to 14, we can use the formula for the variance of the first \( n \) natural numbers.

### Step-by-Step Solution:

1. **Understand the Variance Formula**:
The variance \( \sigma^2 \) of the first \( n \) natural numbers is given by the formula:
\[ \sigma^2 = \frac{n^2 - 1}{12} \]

2. **Set the Variance Equal to 14**:
According to the problem, we have:
\[ \frac{n^2 - 1}{12} = 14 \]

3. **Multiply Both Sides by 12**:
To eliminate the fraction, multiply both sides of the equation by 12:
\[ n^2 - 1 = 14 \times 12 \]

4. **Calculate \( 14 \times 12 \)**:
Calculate the right-hand side:
\[ 14 \times 12 = 168 \]
So, we have:
\[ n^2 - 1 = 168 \]

5. **Add 1 to Both Sides**:
To isolate \( n^2 \), add 1 to both sides:
\[ n^2 = 168 + 1 \]
\[ n^2 = 169 \]

6. **Take the Square Root**:
To find \( n \), take the square root of both sides:
\[ n = \sqrt{169} \]
\[ n = 13 \]

### Final Answer:
Thus, the value of \( n \) is \( 13 \).