Two consecutive number from n natural numbers `1,2,3,……,` n are removed. Arithmetic mean of the remaining numbers is `(105)/(4).`
Let removed numbers are `x _(1), x _(2)` then `x _(1)+ x _(2) + n=`

Options

A. 61
B. 63
C. 65
D. 69

Answer

To solve the problem, we need to find the sum of two consecutive natural numbers removed from the first n natural numbers, given that the arithmetic mean of the remaining numbers is \( \frac{105}{4} \).

### Step-by-Step Solution:

1. **Understanding the Problem**:
We have the first \( n \) natural numbers: \( 1, 2, 3, \ldots, n \). We remove two consecutive numbers \( x_1 \) and \( x_2 \) (where \( x_2 = x_1 + 1 \)). We need to find \( x_1 + x_2 + n \).

2. **Sum of First n Natural Numbers**:
The sum of the first \( n \) natural numbers is given by:
\[ S_n = \frac{n(n + 1)}{2} \]

3. **Sum of the Removed Numbers**:
If we remove \( x_1 \) and \( x_2 \), the sum of the remaining numbers becomes:
\[ S_{\text{remaining}} = S_n - (x_1 + x_2) = \frac{n(n + 1)}{2} - (x_1 + (x_1 + 1)) = \frac{n(n + 1)}{2} - (2x_1 + 1) \]

4. **Number of Remaining Terms**:
After removing two numbers, the count of remaining numbers is \( n - 2 \).

5. **Arithmetic Mean of Remaining Numbers**:
The arithmetic mean of the remaining numbers is given as:
\[ \text{Mean} = \frac{S_{\text{remaining}}}{n - 2} = \frac{\frac{n(n + 1)}{2} - (2x_1 + 1)}{n - 2} \]
Setting this equal to \( \frac{105}{4} \):
\[ \frac{\frac{n(n + 1)}{2} - (2x_1 + 1)}{n - 2} = \frac{105}{4} \]

6. **Cross-Multiplying**:
Cross-multiplying gives:
\[ 4\left(\frac{n(n + 1)}{2} - (2x_1 + 1)\right) = 105(n - 2) \]
Simplifying:
\[ 2n(n + 1) - 8x_1 - 4 = 105n - 210 \]
Rearranging:
\[ 2n^2 + 2n - 105n + 206 - 8x_1 = 0 \]
Which simplifies to:
\[ 2n^2 - 103n + 206 - 8x_1 = 0 \]

7. **Finding Integer Solutions**:
For \( n \) to be an integer, the discriminant of the quadratic must be a perfect square:
\[ D = (-103)^2 - 4 \cdot 2 \cdot (206 - 8x_1) \]
This must yield a perfect square.

8. **Assuming n is Even**:
Let \( n = 2r \). Substituting \( n \) into the equation:
\[ 8r^2 - 206r + 206 - 8x_1 = 0 \]
Solving for \( x_1 \):
\[ 8x_1 = 8r^2 - 206r + 206 \]
\[ x_1 = r^2 - \frac{206}{8}r + \frac{206}{8} \]

9. **Finding Values**:
We can find integer values for \( r \) that satisfy the conditions. After testing values, we find \( n = 50 \) and \( x_1 = 7 \).

10. **Final Calculation**:
Thus, \( x_1 + x_2 + n = 7 + 8 + 50 = 65 \).

### Final Answer:
\[ x_1 + x_2 + n = 65 \]