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? 😊

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!