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The problem asks for the explicit formula of a geometric sequence where the first term is \( g_1 = 5 \) and each subsequent term is defined by the recurrence relation:

\[
g_n = 3 \cdot g_{n-1} \quad \text{for } n \geq 2.
\]

### Step 1: Derive the general formula for \( g_n \)
The formula for a geometric sequence can be written as:
\[
g_n = g_1 \cdot r^{n-1},
\]
where \( g_1 \) is the first term and \( r \) is the common ratio.

Here:
- \( g_1 = 5 \),
- \( r = 3 \) (from the given recurrence relation \( g_n = 3 \cdot g_{n-1} \)).

Substitute these values into the formula:
\[
g_n = 5 \cdot 3^{n-1}.
\]

### Step 2: Match this with the answer choices
The formula \( g_n = 5 \cdot 3^{n-1} \) corresponds to **option D**, which is:
\[
f(x) = 5 \cdot 3^{(n-1)}.
\]

### Final Answer:
**D**: \( f(x) = 5 \cdot 3^{(n-1)} \).

Would you like a step-by-step explanation for how this formula works or have further questions? 😊

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### 5 Follow-up Questions:
1. What is the common ratio in a geometric sequence, and how is it used in the formula?
2. How do you identify the first term and the recurrence relation from a sequence problem?
3. Can geometric sequences also have fractions or negative common ratios?
4. How does the formula \( g_n = g_1 \cdot r^{n-1} \) change for different starting values?
5. How would you find the sum of the first \( n \) terms of this geometric sequence?

### Tip:
When working with recurrence relations, always check if the problem can be simplified into a standard sequence formula!

The first term in a geometric sequence is g1 = 5. For n ≥ 2, the terms of the geometric sequence are given by gn = 3 ⋅ g(n−1). For which of the following functions is gn = f(n) for positive integer values of n?

Learn how to derive the explicit formula for a geometric sequence where the first term is g1 = 5 and gn = 3 ⋅ g(n−1). Includes step-by-step analysis using the geometric sequence formula and recurrence relations. Ideal for high school students (Grades 9-12).

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