Prove that `((n + 1)/(2))^(n) gt n!`

Explanation

To prove that \(\left(\frac{n + 1}{2}\right)^{n} > n!\), 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 \(a_1, a_2, \ldots, a_n\):
\[
\frac{a_1 + a_2 + \ldots + a_n}{n} \geq \sqrt[n]{a_1 a_2 \ldots a_n}
...