A sequence `a_n , n in N` be an A.P. such that `a_7` = 9 and `a_1 a_2 a_7` is least, then the common difference is:

Options

A. `13/20`
B. `23/20`
C. `33/20`
D. `43/20`

Answer

To solve the problem step by step, we will follow the logic outlined in the video transcript while providing a clearer structure.

### Step-by-Step Solution:

1. **Understanding the Arithmetic Progression (A.P.)**:
The nth term of an A.P. can be expressed as:
\[ a_n = a + (n-1)d \]
where \( a \) is the first term and \( d \) is the common difference.

2. **Finding \( a_7 \)**:
Given \( a_7 = 9 \):
\[ a_7 = a + (7-1)d = a + 6d \]
Therefore, we have:
\[ a + 6d = 9 \quad \text{(Equation 1)} \]

3. **Finding \( a_1 \) and \( a_2 \)**:
The first term \( a_1 \) is simply \( a \):
\[ a_1 = a \]
The second term \( a_2 \) is:
\[ a_2 = a + d \quad \text{(Equation 2)} \]

4. **Finding the product \( a_1 a_2 a_7 \)**:
We need to minimize the product:
\[ P = a_1 \cdot a_2 \cdot a_7 = a \cdot (a + d) \cdot 9 \]
Substituting \( a_7 = 9 \):
\[ P = 9a(a + d) \quad \text{(Equation 3)} \]

5. **Substituting \( d \) from Equation 1**:
From Equation 1, we can express \( d \) in terms of \( a \):
\[ 6d = 9 - a \implies d = \frac{9 - a}{6} \]
Substitute \( d \) into Equation 3:
\[ P = 9a\left(a + \frac{9 - a}{6}\right) \]
Simplifying this:
\[ P = 9a\left(\frac{6a + 9 - a}{6}\right) = 9a\left(\frac{5a + 9}{6}\right) \]
\[ P = \frac{9a(5a + 9)}{6} = \frac{15a^2 + 27a}{2} \quad \text{(Equation 4)} \]

6. **Finding the minimum of \( P \)**:
To find the minimum, we take the derivative of \( P \) with respect to \( a \) and set it to zero:
\[ \frac{dP}{da} = \frac{1}{2}(30a + 27) = 0 \]
Solving for \( a \):
\[ 30a + 27 = 0 \implies 30a = -27 \implies a = -\frac{27}{30} = -\frac{9}{10} \]

7. **Finding \( d \)**:
Substitute \( a = -\frac{9}{10} \) back into the equation for \( d \):
\[ d = \frac{9 - a}{6} = \frac{9 - \left(-\frac{9}{10}\right)}{6} = \frac{9 + \frac{9}{10}}{6} = \frac{\frac{90}{10} + \frac{9}{10}}{6} = \frac{\frac{99}{10}}{6} = \frac{99}{60} = \frac{33}{20} \]

### Final Answer:
The common difference \( d \) is:
\[ \boxed{\frac{33}{20}} \]