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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?

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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.

The first five terms of a geometric sequence are shown in the table to the right. Which equation can be used to find the nth term in the sequence?

This solution helps to find the nth term in a geometric sequence with a common ratio of 3. Learn how to identify the correct equation and apply the general formula for geometric sequences, suitable for Grade 9 students.

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