Understanding the Problem

We are given n positive numbers whose product is 1. We need to determine the minimum possible value for their sum.

Applying the AM-GM Inequality

The Arithmetic Mean-Geometric Mean (AM-GM) inequality states that for any set of n non-negative numbers $a_1, a_2, ..., a_n$, the arithmetic mean is greater than or equal to the geometric mean:

$ \frac{a_1 + a_2 + ... + a_n}{n} \ge \sqrt[n]{a_1 a_2 ... a_n} $

Equality holds if and only if all the numbers are equal, i.e., $a_1 = a_2 = ... = a_n$.

Deriving the Sum's Minimum Value

In this problem:

• The numbers $a_1, a_2, ..., a_n$ are positive.
• Their product is given as unity: $a_1 a_2 ... a_n = 1$.

Substitute the product into the AM-GM inequality:

$ \frac{a_1 + a_2 + ... + a_n}{n} \ge \sqrt[n]{1} $

Since $\sqrt[n]{1} = 1$, the inequality becomes:

$ \frac{a_1 + a_2 + ... + a_n}{n} \ge 1 $

Multiply both sides by n (which is a positive integer, so the inequality direction remains unchanged):

$ a_1 + a_2 + ... + a_n \ge n $

Conclusion

The inequality $a_1 + a_2 + ... + a_n \ge n$ shows that the sum of the n positive numbers is always greater than or equal to n. Therefore, the sum is never less than n.

This corresponds to Option 4.