The sum of three terms in A.P. is 33 and their products is 1155. Find the terms.

Answer

To solve the problem, we need to find three terms in an Arithmetic Progression (A.P.) given that their sum is 33 and their product is 1155. Let's denote the three terms as \(a - d\), \(a\), and \(a + d\).

### Step-by-Step Solution:

1. **Set Up the Equations**:
- The sum of the three terms is given by:
\[ (a - d) + a + (a + d) = 33 \]
- Simplifying this, we have:
\[ 3a = 33 \]

2. **Solve for \(a\)**:
- Dividing both sides by 3:
\[ a = \frac{33}{3} = 11 \]

3. **Set Up the Product Equation**:
- The product of the three terms is given by:
\[ (a - d) \cdot a \cdot (a + d) = 1155 \]
- We can simplify the product using the identity \( (a + d)(a - d) = a^2 - d^2 \):
\[ a \cdot (a^2 - d^2) = 1155 \]

4. **Substitute \(a\)**:
- Substituting \(a = 11\) into the product equation:
\[ 11 \cdot (11^2 - d^2) = 1155 \]
- Calculating \(11^2\):
\[ 11 \cdot (121 - d^2) = 1155 \]

5. **Divide by 11**:
- Dividing both sides by 11:
\[ 121 - d^2 = \frac{1155}{11} = 105 \]

6. **Solve for \(d^2\)**:
- Rearranging gives:
\[ d^2 = 121 - 105 = 16 \]

7. **Find \(d\)**:
- Taking the square root:
\[ d = \sqrt{16} = 4 \quad \text{or} \quad d = -4 \]

8. **Find the Terms**:
- When \(d = 4\):
- The terms are:
\[ a - d = 11 - 4 = 7, \quad a = 11, \quad a + d = 11 + 4 = 15 \]
- So the terms are \(7, 11, 15\).

- When \(d = -4\):
- The terms are:
\[ a - d = 11 - (-4) = 15, \quad a = 11, \quad a + d = 11 + (-4) = 7 \]
- So the terms are \(15, 11, 7\).

### Final Answer:
The three terms in A.P. are \(7, 11, 15\) (or \(15, 11, 7\) depending on the order).