Let's analyze the question and solve it.

The problem is about finding the formula for the $n$-th term ($U_n$) of a geometric sequence. We are given a sequence:

$2, 8, 32, 128, \dots$

Step-by-Step Solution:

1. Identify the first term ($a$):

   • From the sequence, the first term $a = 2$.

2. Find the common ratio ($r$):

   • To find the common ratio, divide the second term by the first term: $r = \frac{8}{2} = 4$

3. General formula for the $n$-th term of a geometric sequence:

   • The formula for the $n$-th term of a geometric sequence is: $U_n = a \cdot r^{n-1}$
   • Substituting $a = 2$ and $r = 4$: $U_n = 2 \cdot 4^{n-1}$

Final Answer:

The formula for the $n$-th term of the sequence is: $U_n = 2 \cdot 4^{n-1}$

Would you like a further breakdown or have any questions about this solution?

Related Questions:

1. How would the formula change if the first term were different, for example $a = 3$?
2. What happens to the $n$-th term formula if the common ratio is negative?
3. Can we derive the sum formula for the first $n$ terms of this geometric sequence?
4. How would we find the term number $n$ if we know the value of a specific term, say $U_n = 512$?
5. What would be the $n$-th term formula if this was an arithmetic sequence instead of a geometric one?

Tip:

When finding the common ratio in a geometric sequence, always ensure it’s consistent by dividing successive terms.