Consider the A.P. `50,48,46,44 ……. If S _(n)` denotes the sum to n terms of this A.P. then

Options

A. `S_(n) ` is maximum for n =25`
B. the first negative terms is `26^(th)` term
C. the first negative term is `27 ^(th)` term
D. the maximum value of `S_(n)` is 650

Answer

To solve the problem step by step, we will analyze the given arithmetic progression (A.P.) and calculate the required values.

### Step 1: Identify the first term (A) and common difference (D)
The given A.P. is:
\[ 50, 48, 46, 44, \ldots \]

- The first term \( A = 50 \)
- The common difference \( D = 48 - 50 = -2 \)

### Step 2: Find the first term that is zero
To find the first term that is zero, we can use the formula for the n-th term of an A.P.:
\[ T_n = A + (n - 1)D \]

Setting \( T_n = 0 \):
\[ 0 = 50 + (n - 1)(-2) \]

### Step 3: Solve for n
Rearranging the equation:
\[ 50 - 2(n - 1) = 0 \]
\[ 50 = 2(n - 1) \]
\[ 25 = n - 1 \]
\[ n = 26 \]

Thus, the first term that is zero is the 26th term.

### Step 4: Determine the first negative term
Since the 26th term is zero, the first negative term will be the 27th term:
\[ T_{27} = 50 + (27 - 1)(-2) \]
\[ T_{27} = 50 + 26(-2) \]
\[ T_{27} = 50 - 52 = -2 \]

So, the first negative term is the 27th term.

### Step 5: Find the maximum sum \( S_n \)
The sum of the first n terms of an A.P. is given by:
\[ S_n = \frac{n}{2} \left( 2A + (n - 1)D \right) \]

To find when \( S_n \) is maximum, we note that since the 26th term is zero, the sum will be maximum at \( n = 25 \).

### Step 6: Calculate \( S_{25} \)
Substituting \( n = 25 \):
\[ S_{25} = \frac{25}{2} \left( 2 \times 50 + (25 - 1)(-2) \right) \]
\[ S_{25} = \frac{25}{2} \left( 100 - 48 \right) \]
\[ S_{25} = \frac{25}{2} \times 52 \]
\[ S_{25} = 25 \times 26 = 650 \]

### Conclusion
1. The first negative term is the 27th term.
2. The sum \( S_n \) is maximum for \( n = 25 \).
3. The maximum value of \( S_{25} \) is 650.

### Final Options
- Option 1: \( S_n \) is maximum for \( n = 25 \) (Correct)
- Option 2: The first negative term is the 26th term (Incorrect)
- Option 3: The first negative term is the 27th term (Correct)
- Option 4: The maximum value of \( S_n \) is 650 (Correct)

Thus, the correct options are 1, 3, and 4.