If the sum and product of the first three terms in an AP are 33 and 1155, respectively, then a value of its 11th term is

Options

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

Explanation

To solve the problem, we need to find the 11th term of an arithmetic progression (AP) given the sum and product of the first three terms.

### Step-by-Step Solution:

1. **Define the first three terms of the AP**:
Let the first three terms of the AP be \( a - d, a, a + d \).

2. **Use the sum of the terms**:
The sum of the first three terms is given as 33:
\[ (a - d) + a + (a + d) = 3a = 33 \]
From this, we can solve for \( a \):
\[ 3a = 33 \implies a = 11 \]

3. **Use the product of the terms**:
The product of the first three terms is given as 1155:
\[ (a - d) \cdot a \cdot (a + d) = 1155 \]
Substituting \( a = 11 \):
\[ (11 - d) \cdot 11 \cdot (11 + d) = 1155 \]
This simplifies to:
\[ 11 \cdot (11^2 - d^2) = 1155 \]
\[ 11 \cdot (121 - d^2) = 1155 \]
Dividing both sides by 11:
\[ 121 - d^2 = 105 \]
Rearranging gives:
\[ d^2 = 121 - 105 = 16 \]
Taking the square root:
\[ d = 4 \quad \text{or} \quad d = -4 \]

4. **Find the 11th term**:
The 11th term of the AP is given by:
\[ a + 9d \]
We will calculate this for both values of \( d \).

- For \( d = 4 \):
\[ a + 9d = 11 + 9 \cdot 4 = 11 + 36 = 47 \]

- For \( d = -4 \):
\[ a + 9d = 11 + 9 \cdot (-4) = 11 - 36 = -25 \]

5. **Conclusion**:
The possible values for the 11th term are 47 and -25. The question asks for a value of the 11th term, so we can state:
\[ \text{A value of the 11th term is } -25. \]