Sum of three terms of an A.P. is 33 and their product is 792. The least of them is

Options

A. 7
B. 11
C. 18
D. none of these

Answer

To solve the problem, we need to find the least of three terms in an arithmetic progression (A.P.) given that their sum is 33 and their product is 792.

### Step-by-Step Solution:

1. **Define the Terms**:
Let the three terms of the A.P. be:
- First term: \( a - d \)
- Second term: \( a \)
- Third term: \( a + d \)

2. **Set Up the Sum Equation**:
According to the problem, the sum of these three terms is given by:
\[ (a - d) + a + (a + d) = 3a = 33 \]
From this equation, we can solve for \( a \):
\[ 3a = 33 \implies a = \frac{33}{3} = 11 \]

3. **Set Up the Product Equation**:
The product of the three terms is given by:
\[ (a - d) \cdot a \cdot (a + d) = 792 \]
Substituting \( a = 11 \):
\[ (11 - d) \cdot 11 \cdot (11 + d) = 792 \]

4. **Simplify the Product Equation**:
This can be rewritten as:
\[ 11 \cdot (11^2 - d^2) = 792 \]
Simplifying further:
\[ 11 \cdot (121 - d^2) = 792 \]
Dividing both sides by 11:
\[ 121 - d^2 = \frac{792}{11} = 72 \]

5. **Solve for \( d^2 \)**:
Rearranging gives:
\[ d^2 = 121 - 72 = 49 \]
Taking the square root:
\[ d = 7 \quad (\text{since } d \text{ is a distance, we take the positive root}) \]

6. **Find the Three Terms**:
Now we can find the three terms:
- First term: \( a - d = 11 - 7 = 4 \)
- Second term: \( a = 11 \)
- Third term: \( a + d = 11 + 7 = 18 \)

7. **Identify the Least Term**:
The three terms are \( 4, 11, \) and \( 18 \). Therefore, the least of these terms is:
\[ \text{Least term} = 4 \]

### Final Answer:
The least of the three terms is **4**.