The question provides the first five terms of a geometric sequence: $g_1 = 1$, $g_2 = 3$, $g_3 = 9$, $g_4 = 27$, and $g_5 = 81$. To find the equation for the $n$-th term, $g_n$, we need to determine the common ratio and the initial term.

1. Identify the common ratio: Each term is obtained by multiplying the previous term by 3 (e.g., $1 \times 3 = 3$, $3 \times 3 = 9$, etc.). So, the common ratio $r = 3$.

2. Use the formula for a geometric sequence: The general formula for the $n$-th term of a geometric sequence is: $g_n = g_1 \cdot r^{n-1}$ where $g_1 = 1$ and $r = 3$.

3. Substitute the values: $g_n = 1 \cdot 3^{n-1}$

From the answer choices, option C: $g_n = 1 \cdot 3^{n-1}$ is correct.

Would you like further explanation or have any questions?

Here are five related questions:

1. What would the 6th term of this sequence be?
2. How would the formula change if the initial term were 2 instead of 1?
3. What is the sum of the first 5 terms of this sequence?
4. How can you determine the common ratio in a geometric sequence?
5. What distinguishes a geometric sequence from an arithmetic sequence?

Tip: In a geometric sequence, each term is found by multiplying the previous term by a constant, known as the common ratio.