Statement-1 : If `a_1,a_2,a_3`,….. `a_n` are positive real numbers , whose product is a fixed number c, then the minimum value of `a_1+a_2+…. + a_(n-1)+2a_n` is `n(2C)^(1/n)`
Statement-2 : A.M. `ge` G.M.

Options

A. a) Both statement I and statement II are correct. Statement II is the correct explanation of statement I.
B. b) Both statement I and statement II are correct. Statement II is not the correct explanation of statement I.
C. c) Statement I is correct and statement II is incorrect.
D. d) Statement I is incorrect and statement II is correct.

Answer

To solve the problem, we need to find the minimum value of the expression \( a_1 + a_2 + a_3 + \ldots + a_{n-1} + 2a_n \) given that the product of the positive real numbers \( a_1, a_2, \ldots, a_n \) is a fixed number \( c \).

### Step-by-Step Solution:

1. **Understand the Given Information**:
We are given that \( a_1, a_2, \ldots, a_n \) are positive real numbers such that:
\[ a_1 \cdot a_2 \cdot a_3 \cdots a_n = c \]

2. **Rewrite the Expression**:
We need to minimize the expression:
\[ S = a_1 + a_2 + a_3 + \ldots + a_{n-1} + 2a_n \]

3. **Use the AM-GM Inequality**:
According to the Arithmetic Mean-Geometric Mean (AM-GM) inequality, we have:
\[ \text{AM} \geq \text{GM} \]
For our case, we can apply AM-GM to the terms \( a_1, a_2, \ldots, a_{n-1}, 2a_n \).

4. **Calculate the Arithmetic Mean**:
The arithmetic mean of these \( n \) terms is:
\[ \text{AM} = \frac{a_1 + a_2 + \ldots + a_{n-1} + 2a_n}{n} \]

5. **Calculate the Geometric Mean**:
The geometric mean of these terms is:
\[ \text{GM} = (a_1 \cdot a_2 \cdots a_{n-1} \cdot (2a_n))^{1/n} \]
This can be rewritten as:
\[ \text{GM} = (2 \cdot a_1 \cdot a_2 \cdots a_{n-1} \cdot a_n)^{1/n} = (2c)^{1/n} \]

6. **Apply the AM-GM Inequality**:
By the AM-GM inequality:
\[ \frac{a_1 + a_2 + \ldots + a_{n-1} + 2a_n}{n} \geq (2c)^{1/n} \]
Therefore, we can multiply both sides by \( n \):
\[ a_1 + a_2 + \ldots + a_{n-1} + 2a_n \geq n(2c)^{1/n} \]

7. **Conclusion**:
The minimum value of \( S = a_1 + a_2 + \ldots + a_{n-1} + 2a_n \) is:
\[ n(2c)^{1/n} \]
Hence, Statement-1 is correct.

8. **Statement-2 Validation**:
Statement-2 asserts that AM ≥ GM is a true statement, which is indeed correct.

### Final Answer:
Both Statement 1 and Statement 2 are correct, and Statement 2 is the correct explanation of Statement 1.