If `a_(i) gt 0 AA I in N` such that `prod_(i=1)^(n) a_(i) = 1`, then prove that `(1 + a_(1)) (1 + a_(2)) (1 + a_(3)) .... (1 + a_(n)) ge 2^(n)`

Explanation

To prove that \((1 + a_1)(1 + a_2)(1 + a_3) \ldots (1 + a_n) \geq 2^n\) given that \(a_i > 0\) for all \(i \in \mathbb{N}\) and \(\prod_{i=1}^{n} a_i = 1\), we can use the Arithmetic Mean-Geometric Mean (AM-GM) inequality.

### Step-by-Step Solution:

1. **Understanding the AM-GM Inequality**:
The AM-GM inequality states that 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}
...