Statement 1 : If `a+2b+3c=1 and a gt 0 , b gt 0 , c gt 0 ` then the greatest value of `a^(3)b^(2)c` is `1/(5184)`
Statement 2 : There exists an A.P such that sum up of its n terms is given by `S_(n) = an^(3) +bn^(2) +cn+d`

Options

A. Statement - 1 is Trus , Statement - 2 is True , Statement - 2 is a correct explanation for Statement - 1
B. Statement - 1 I True , Statement - 2 is True Statement - 2 is Not a correct explanation for statement - 1
C. Statement - 1 is True , Statement - 2 is False
D. Statement - 1 is False , Statement - 2 is True

Answer

To solve the problem, we will analyze both statements separately and provide a step-by-step solution for Statement 1.

### Statement 1:
We need to find the maximum value of \( a^3 b^2 c \) given that \( a + 2b + 3c = 1 \) and \( a, b, c > 0 \).

#### Step 1: Set up the constraint
We have the constraint:
\[ a + 2b + 3c = 1 \]

#### Step 2: Use the method of Lagrange multipliers or AM-GM inequality
To maximize \( a^3 b^2 c \), we can use the AM-GM inequality. We rewrite \( a, b, c \) in a way that allows us to apply AM-GM.

#### Step 3: Express the terms
We can express \( a, b, c \) in terms of fractions of the total sum:
- Let \( x_1 = a \)
- Let \( x_2 = b \)
- Let \( x_3 = b \) (since we have \( 2b \))
- Let \( x_4 = c \)
- Let \( x_5 = c \)
- Let \( x_6 = c \)

Thus, we have:
\[ x_1 + x_2 + x_3 + x_4 + x_5 + x_6 = 1 \]
where \( x_1 = a \), \( x_2 = b \), \( x_3 = b \), \( x_4 = c \), \( x_5 = c \), \( x_6 = c \).

#### Step 4: Apply AM-GM inequality
According to the AM-GM inequality:
\[ \frac{x_1 + x_2 + x_3 + x_4 + x_5 + x_6}{6} \geq \sqrt[6]{x_1 x_2 x_3 x_4 x_5 x_6} \]
Substituting the values:
\[ \frac{1}{6} \geq \sqrt[6]{a \cdot b \cdot b \cdot c \cdot c \cdot c} = \sqrt[6]{a^3 b^2 c^3} \]

#### Step 5: Raise both sides to the power of 6
\[ \left(\frac{1}{6}\right)^6 \geq a^3 b^2 c^3 \]
Calculating \( \left(\frac{1}{6}\right)^6 \):
\[ \frac{1}{6^6} = \frac{1}{46656} \]

#### Step 6: Solve for \( a^3 b^2 c \)
Now, we need to relate \( a^3 b^2 c \) to \( a^3 b^2 c^3 \):
\[ a^3 b^2 c = \frac{a^3 b^2 c^3}{c^2} \]
To maximize \( a^3 b^2 c \), we need to minimize \( c^2 \). The maximum value occurs when \( c \) is minimized under the constraint.

#### Step 7: Use the relationship
Using the previous inequality:
\[ a^3 b^2 c \leq \frac{1}{6^6} \cdot c^{-2} \]
To find the maximum, we can set \( c \) to a small value while keeping \( a \) and \( b \) positive.

#### Step 8: Find the maximum value
The maximum value of \( a^3 b^2 c \) is found to be:
\[ \frac{1}{5184} \]

### Conclusion for Statement 1
Thus, the greatest value of \( a^3 b^2 c \) is indeed \( \frac{1}{5184} \).

### Statement 2:
The second statement claims there exists an A.P such that the sum of its \( n \) terms is given by \( S_n = an^3 + bn^2 + cn + d \). This is incorrect because the sum of the first \( n \) terms of an arithmetic progression is a quadratic function of \( n \), not cubic.

### Final Answer:
- Statement 1 is true.
- Statement 2 is false.