The given problem describes an arithmetic sequence $(u_n)$, with the first term $u_1 = -3$ and the common difference $d = \frac{1}{2}$. We are tasked with finding the correct formula for the general term $u_n$.

The general formula for the $n$-th term of an arithmetic sequence is:

$u_n = u_1 + (n - 1) \cdot d$

Substituting the given values:

$u_n = -3 + (n - 1) \cdot \frac{1}{2}$

Now, let's simplify this:

$u_n = -3 + \frac{n - 1}{2}$

This matches option C in the provided choices.

So, the correct answer is C: $u_n = -3 + \frac{1}{2} \cdot (n - 1)$

Would you like more details or further clarifications?

Here are five related questions you could explore:

1. What is the sum of the first 10 terms of this arithmetic sequence?
2. How do you derive the formula for the $n$-th term of an arithmetic sequence?
3. What is the difference between an arithmetic sequence and a geometric sequence?
4. How does changing the common difference $d$ affect the growth of the sequence?
5. Can you graph the first few terms of this arithmetic sequence to visualize its behavior?

Tip: When dealing with arithmetic sequences, always check the formula for the $n$-th term carefully to ensure proper placement of constants and variables.