Two consecutive numbers from 1,2,3 …., n are removed .The arithmetic mean of the remaining numbers is 105/4
The sum of all numbers

Options

A. `S_(40)=-820`
B. `S_(2n) gt S_(2n+2)`
C. `S_(51)=1326`
D. `S_(2n +1) gt S_(2n-1)`

Explanation

To solve the problem step by step, we need to find the sum of all numbers from 1 to n after removing two consecutive numbers, given that the arithmetic mean of the remaining numbers is \( \frac{105}{4} \).

### Step 1: Understand the Problem
We are given that two consecutive numbers \( a \) and \( a+1 \) are removed from the set \( \{1, 2, 3, \ldots, n\} \). We need to find the sum of all numbers from 1 to n.

### Step 2: Calculate the Sum of All Numbers
The sum of the first \( n \) natural numbers is given by the formula:
\[ S_n = \frac{n(n+1)}{2} \]

### Step 3: Calculate the Sum After Removing Two Numbers
When we remove the two consecutive numbers \( a \) and \( a+1 \), the sum of the remaining numbers becomes:
\[ S' = S_n - (a + (a + 1)) = S_n - (2a + 1) \]
Substituting \( S_n \):
\[ S' = \frac{n(n+1)}{2} - (2a + 1) \]

### Step 4: Find the Number of Remaining Elements
After removing two numbers, the number of remaining elements is:
\[ n - 2 \]

### Step 5: Set Up the Equation for Arithmetic Mean
The arithmetic mean of the remaining numbers is given by:
\[ \text{Mean} = \frac{S'}{n - 2} \]
Setting this equal to \( \frac{105}{4} \):
\[ \frac{\frac{n(n+1)}{2} - (2a + 1)}{n - 2} = \frac{105}{4} \]

### Step 6: Cross Multiply to Eliminate the Fraction
Cross multiplying gives:
\[ 4 \left( \frac{n(n+1)}{2} - (2a + 1) \right) = 105(n - 2) \]
This simplifies to:
\[ 2n(n+1) - 4(2a + 1) = 105n - 210 \]

### Step 7: Rearranging the Equation
Rearranging the equation yields:
\[ 2n^2 + 2n - 105n + 210 - 8a - 4 = 0 \]
This simplifies to:
\[ 2n^2 - 103n + 206 - 8a = 0 \]

### Step 8: Solve for n in Terms of a
Rearranging gives:
\[ 2n^2 - 103n + (206 - 8a) = 0 \]
Using the quadratic formula \( n = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
\[ n = \frac{103 \pm \sqrt{(-103)^2 - 4 \cdot 2 \cdot (206 - 8a)}}{2 \cdot 2} \]

### Step 9: Determine Conditions for n
Since \( n \) must be a positive integer, we can analyze the discriminant \( b^2 - 4ac \) to ensure it is a perfect square.

### Step 10: Find Values of a and n
After testing integer values for \( a \) and ensuring \( n \) is even, we can find suitable pairs.

### Step 11: Calculate the Final Sum
Once we have \( n \), we can calculate the sum:
\[ S_n = \frac{n(n+1)}{2} \]
For \( n = 50 \):
\[ S_n = \frac{50 \cdot 51}{2} = 1275 \]

### Final Answer
The sum of all numbers from 1 to \( n \) is \( 1275 \).

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