If `a_1,a_2 ` …. `a_n` are positive real numbers whose product is a fixed real number c, then the minimum value of `1+a_1 +a_2` +….. + `a_(n-1) + a_n` is :

Options

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

Answer

To solve the problem, we need to find the minimum value of the expression \( S = 1 + a_1 + a_2 + \ldots + a_n \) given that the product \( a_1 \cdot a_2 \cdots a_n = c \), where \( c \) is a fixed positive real number.

### Step-by-Step Solution:

1. **Understanding the Problem**:
We have \( n \) positive real numbers \( a_1, a_2, \ldots, a_n \) such that their product is a constant \( c \). We want to minimize the sum \( S = 1 + a_1 + a_2 + \ldots + a_n \).

2. **Using the AM-GM Inequality**:
The Arithmetic Mean-Geometric Mean (AM-GM) inequality states that for any set of positive real numbers, the arithmetic mean is greater than or equal to the geometric mean. Therefore, we can apply this inequality to our case.

\[ \frac{1 + a_1 + a_2 + \ldots + a_n}{n + 1} \geq (1 \cdot a_1 \cdot a_2 \cdots a_n)^{\frac{1}{n + 1}} \]

3. **Substituting the Product**:
Since the product \( a_1 \cdot a_2 \cdots a_n = c \), we can substitute this into the inequality:

\[ \frac{1 + a_1 + a_2 + \ldots + a_n}{n + 1} \geq (1 \cdot c)^{\frac{1}{n + 1}} = c^{\frac{1}{n + 1}} \]

4. **Rearranging the Inequality**:
Now, we can rearrange the inequality to express \( S \):

\[ 1 + a_1 + a_2 + \ldots + a_n \geq (n + 1) c^{\frac{1}{n + 1}} \]

5. **Finding the Minimum Value**:
The minimum value of \( S \) occurs when the equality condition of the AM-GM inequality is satisfied, which happens when all the terms are equal. Thus, we set:

\[ 1 = a_1 = a_2 = \ldots = a_n \]

Let \( a_1 = a_2 = \ldots = a_n = x \). Then:

\[ x^n = c \implies x = c^{\frac{1}{n}} \]

Therefore, substituting back into \( S \):

\[ S = 1 + n \cdot c^{\frac{1}{n}} = 1 + n \cdot c^{\frac{1}{n}} \]

However, to find the minimum value of \( S \) in terms of \( c \):

\[ S = (n + 1) c^{\frac{1}{n + 1}} \]

### Final Answer:
Thus, the minimum value of \( S = 1 + a_1 + a_2 + \ldots + a_n \) is:

\[ \boxed{(n + 1) c^{\frac{1}{n + 1}}} \]