A teacher gave sum to his class to find the average of n numbers viz. 1,2,3,4,5,6,… etc. but when the teacher checked the solution, he has found that during the calculation a student just missed a number for the addition thus his average of the n number was 15. The value of n is :

Options

A. 30
B. 26
C. 31
D. not unique

Answer

To solve the problem, we need to find the value of \( n \) given that the average of the first \( n \) natural numbers (1, 2, 3, ..., n) was calculated incorrectly because one number was missed, resulting in an average of 15.

### Step-by-Step Solution:

1. **Understand the Average Formula**:
The average of the first \( n \) natural numbers can be calculated using the formula:
\[ \text{Average} = \frac{\text{Sum of first } n \text{ natural numbers}}{n} = \frac{\frac{n(n+1)}{2}}{n} = \frac{n+1}{2} \]

2. **Set Up the Equation**:
Since the student calculated the average as 15, we can set up the equation:
\[ \frac{n+1}{2} = 15 \]

3. **Solve for \( n \)**:
To find \( n \), we first multiply both sides of the equation by 2:
\[ n + 1 = 30 \]
Next, subtract 1 from both sides:
\[ n = 29 \]

4. **Verify the Calculation**:
Now, we need to verify if the average calculated with \( n = 29 \) and missing one number gives the average of 15. The sum of the first 29 natural numbers is:
\[ \text{Sum} = \frac{29 \times 30}{2} = 435 \]
If one number (let's denote it as \( x \)) is missed, the new sum becomes \( 435 - x \), and the new count of numbers is \( 28 \). The new average is:
\[ \text{New Average} = \frac{435 - x}{28} = 15 \]
Multiplying both sides by 28 gives:
\[ 435 - x = 420 \]
Solving for \( x \):
\[ x = 435 - 420 = 15 \]
Since 15 is indeed one of the numbers in the first 29 natural numbers, our calculations are correct.

5. **Conclusion**:
Therefore, the value of \( n \) is:
\[ \boxed{29} \]