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

Options

A. `n(2c)^(1//n)`
B. `(n+1)c^(1//n)`
C. `2nc^(1//n)`
D. `(n+1)(2c)^(1//n)`

Answer

To find the minimum value of \( S = a_1 + a_2 + \ldots + a_{n-1} + 2a_n \) given that the product \( a_1 a_2 \ldots a_n = c \) (where \( c \) is a fixed positive number), we can use the Arithmetic Mean-Geometric Mean (AM-GM) inequality.

### Step-by-Step Solution:

1. **Identify the Terms**:
We have \( n \) terms in total: \( a_1, a_2, \ldots, a_{n-1}, 2a_n \).

2. **Apply the AM-GM Inequality**:
According to the AM-GM inequality, for any non-negative real numbers \( x_1, x_2, \ldots, x_n \),
\[ \frac{x_1 + x_2 + \ldots + x_n}{n} \geq \sqrt[n]{x_1 x_2 \ldots x_n} \]
Here, we can set \( x_1 = a_1, x_2 = a_2, \ldots, x_{n-1} = a_{n-1}, x_n = 2a_n \).

3. **Calculate the Arithmetic Mean**:
The arithmetic mean of our terms is:
\[ \frac{a_1 + a_2 + \ldots + a_{n-1} + 2a_n}{n} \]

4. **Calculate the Geometric Mean**:
The geometric mean is:
\[ \sqrt[n]{a_1 \cdot a_2 \cdots a_{n-1} \cdot (2a_n)} = \sqrt[n]{(a_1 a_2 \cdots a_{n-1}) \cdot 2a_n} \]

5. **Express the Product**:
We know that \( a_1 a_2 \cdots a_n = c \). Therefore, we can express \( a_1 a_2 \cdots a_{n-1} \) as \( \frac{c}{a_n} \).

6. **Substituting into the Geometric Mean**:
Now substituting this into the geometric mean:
\[ \sqrt[n]{\frac{c}{a_n} \cdot 2a_n} = \sqrt[n]{2c} \]

7. **Combine the Inequalities**:
From the AM-GM inequality, we have:
\[ \frac{a_1 + a_2 + \ldots + a_{n-1} + 2a_n}{n} \geq \sqrt[n]{2c} \]
Multiplying both sides by \( n \):
\[ a_1 + a_2 + \ldots + a_{n-1} + 2a_n \geq n \sqrt[n]{2c} \]

8. **Conclusion**:
Thus, the minimum value of \( S = a_1 + a_2 + \ldots + a_{n-1} + 2a_n \) is:
\[ S_{\text{min}} = n \sqrt[n]{2c} \]

### Final Answer:
The minimum value of \( a_1 + a_2 + \ldots + a_{n-1} + 2a_n \) is \( n \sqrt[n]{2c} \).