The sequence given is $4, 2, 1, \dots$, and we are tasked with finding the explicit formula for the $n$-th term of the sequence.

Step-by-Step Solution:

1. Observe the Pattern:

   • The first term ($a_1$) is 4.
   • The second term ($a_2$) is 2.
   • The third term ($a_3$) is 1.

   Each term is halved compared to the previous term. Hence, this is a geometric sequence where the common ratio ($r$) is $\frac{1}{2}$.

2. General Formula for a Geometric Sequence: $a_n = a_1 \cdot r^{n-1}$ Substituting $a_1 = 4$ and $r = \frac{1}{2}$: $a_n = 4 \cdot \left( \frac{1}{2} \right)^{n-1}$

3. Match the Formula with the Options: Among the given choices, the correct formula is: $a_n = 4 \cdot \left( \frac{1}{2} \right)^{n-1}$

   This corresponds to the second option.

Final Answer:

$a_n = 4 \cdot \left( \frac{1}{2} \right)^{n-1}$

Let me know if you need further clarification or have any questions! Here are some related topics to explore:

1. What defines a geometric sequence, and how is it different from arithmetic sequences?
2. How do you derive the general formula for the $n$-th term of a geometric sequence?
3. What happens to the formula if the sequence alternates signs (e.g., 4, -2, 1, ...)?
4. How is the sum of a geometric series calculated?
5. What real-world problems can be modeled using geometric sequences?

Tip: Always check the common ratio in a sequence to determine if it's geometric!