The problem is to identify the pattern in the given sequence and find a general expression for the $n$-th term. The sequence is $1, \frac{2}{2^{2-1}}, \frac{3}{3^{2^2}}, \frac{4}{4^{2^3}}, ...$

1. Problem Description

The problem is to identify the pattern in the given sequence and find a general expression for the $n$-th term. The sequence is $1, \frac{2}{2^{2-1}}, \frac{3}{3^{2^2}}, \frac{4}{4^{2^3}}, ...$

2. Solution Steps

Let's analyze the sequence:

- The first term is $1 = \frac{1}{1} = \frac{1}{1^{2^0}}$

- The second term is $\frac{2}{2^{2-1}} = \frac{2}{2^{1}} = \frac{2}{2^{2^0}}$

- The third term is $\frac{3}{3^{2^2}} = \frac{3}{3^{4}}$

- The fourth term is $\frac{4}{4^{2^3}} = \frac{4}{4^{8}}$

We can observe a pattern here. The $n$-th term of the sequence can be written as:

$a_n = \frac{n}{n^{2^{n-1}}}$ for $n \ge 1$.

3. Final Answer

The general expression for the $n$-th term of the sequence is $a_n = \frac{n}{n^{2^{n-1}}}$.