If the sum of first `3` terms of an `A.P.` is `33` and their product is `1155`. Then the `11^(th)` term of the `A.P`. Is

Options

A. -35
B. 25
C. 36
D. -25

Answer

To solve the problem step by step, we will derive the values of the first three terms of the arithmetic progression (A.P.) and then find the 11th term.

### Step 1: Define the terms of the A.P.
Let the first three terms of the A.P. be \( a - r \), \( a \), and \( a + r \), where \( a \) is the first term and \( r \) is the common difference.

### Step 2: Set up the equation for the sum of the first three terms.
According to the problem, the sum of the first three terms is given as:
\[ (a - r) + a + (a + r) = 33 \]
This simplifies to:
\[ 3a = 33 \]
Thus, we can solve for \( a \):
\[ a = \frac{33}{3} = 11 \]

### Step 3: Set up the equation for the product of the first three terms.
The product of the first three terms is given as:
\[ (a - r) \cdot a \cdot (a + r) = 1155 \]
Substituting \( a = 11 \) into the equation gives:
\[ (11 - r) \cdot 11 \cdot (11 + r) = 1155 \]
This can be rewritten as:
\[ 11 \cdot ((11 - r)(11 + r)) = 1155 \]
Using the difference of squares, we have:
\[ 11 \cdot (121 - r^2) = 1155 \]
Dividing both sides by 11:
\[ 121 - r^2 = 105 \]

### Step 4: Solve for \( r^2 \).
Rearranging the equation gives:
\[ r^2 = 121 - 105 = 16 \]
Taking the square root, we find:
\[ r = \pm 4 \]

### Step 5: Determine the three terms of the A.P.
1. If \( r = 4 \):
- The terms are \( 11 - 4 = 7 \), \( 11 \), and \( 11 + 4 = 15 \).
2. If \( r = -4 \):
- The terms are \( 11 + 4 = 15 \), \( 11 \), and \( 11 - 4 = 7 \).

In both cases, the three terms of the A.P. are \( 7, 11, 15 \).

### Step 6: Find the 11th term of the A.P.
The formula for the \( n \)-th term of an A.P. is given by:
\[ T_n = a + (n - 1) \cdot d \]
Here, \( a = 7 \) (first term) and \( d = 4 \) (common difference). For \( n = 11 \):
\[ T_{11} = 7 + (11 - 1) \cdot 4 \]
Calculating this:
\[ T_{11} = 7 + 10 \cdot 4 = 7 + 40 = 47 \]

### Step 7: Check the case when \( r = -4 \).
Using \( r = -4 \), the terms are \( 15, 11, 7 \):
\[ T_{11} = 15 + (11 - 1)(-4) \]
Calculating this:
\[ T_{11} = 15 + 10 \cdot (-4) = 15 - 40 = -25 \]

### Conclusion
Thus, the two possible values for the 11th term of the A.P. are \( 47 \) and \( -25 \). According to the options, the correct answer is \( -25 \).