Two consecutive number from n natural numbers `1,2,3,……,` n are removed. Arithmetic mean of the remaining numbers is `(105)/(4).`
The value of `n` is:

Options

A. 48
B. 50
C. 52
D. 49

Answer

To solve the problem, we need to find the value of \( n \) given that two consecutive natural numbers are removed from the sequence \( 1, 2, 3, \ldots, n \) and the arithmetic mean of the remaining numbers is \( \frac{105}{4} \).

### Step-by-Step Solution:

1. **Define the Removed Numbers**:
Let the two consecutive numbers removed be \( p \) and \( p+1 \).

2. **Calculate the Sum of First \( n \) Natural Numbers**:
The sum of the first \( n \) natural numbers can be calculated using the formula:
\[ S_n = \frac{n(n+1)}{2} \]

3. **Calculate the Sum After Removing \( p \) and \( p+1 \)**:
The sum of the remaining numbers after removing \( p \) and \( p+1 \) is:
\[ S_{\text{remaining}} = S_n - (p + (p + 1)) = S_n - (2p + 1) \]
Substituting the formula for \( S_n \):
\[ S_{\text{remaining}} = \frac{n(n+1)}{2} - (2p + 1) \]

4. **Calculate the Number of Remaining Terms**:
Since we removed 2 numbers from \( n \), the number of remaining terms is:
\[ n - 2 \]

5. **Set Up the Equation for Arithmetic Mean**:
The arithmetic mean of the remaining numbers is given by:
\[ \text{Arithmetic Mean} = \frac{S_{\text{remaining}}}{n - 2} \]
We know this equals \( \frac{105}{4} \):
\[ \frac{\frac{n(n+1)}{2} - (2p + 1)}{n - 2} = \frac{105}{4} \]

6. **Cross-Multiply to Eliminate the Fraction**:
Cross-multiplying gives:
\[ 4\left(\frac{n(n+1)}{2} - (2p + 1)\right) = 105(n - 2) \]
Simplifying this:
\[ 2n(n + 1) - 4(2p + 1) = 105n - 210 \]

7. **Rearranging the Equation**:
Rearranging gives:
\[ 2n^2 + 2n - 105n + 210 - 8p - 4 = 0 \]
Simplifying further:
\[ 2n^2 - 103n - 8p + 206 = 0 \]

8. **Analyzing the Equation**:
This is a quadratic equation in \( n \). For \( n \) to be a natural number, the discriminant must be a perfect square:
\[ D = b^2 - 4ac = (-103)^2 - 4 \cdot 2 \cdot (206 - 8p) \]
This simplifies to:
\[ D = 10609 - 8p \]

9. **Finding Suitable Values for \( p \)**:
To ensure \( D \) is a perfect square, \( 10609 - 8p \) must be non-negative and a perfect square.

10. **Testing Values**:
We can test integer values for \( p \) to find suitable \( n \). After testing values, we find:
- If \( p = 6 \), then \( n = 50 \) satisfies the equation.

### Conclusion:
The value of \( n \) is:
\[ \boxed{50} \]