1. **State the problem:** We have a geometric sequence with first term $g_1 = 5$ and for $n \geq 2$, $g_n = 3 \cdot g_{n-1}$. We want to find the function $f(n)$ that matches $g_n$ for positive integers $n$. 2. **Recall the formula for geometric sequences:** The $n$th term of a geometric sequence is given by $$g_n = g_1 \cdot r^{n-1}$$ where $r$ is the common ratio. 3. **Identify the common ratio:** From the problem, $g_n = 3 \cdot g_{n-1}$, so the common ratio is $r = 3$. 4. **Write the explicit formula:** Using the formula, $$g_n = 5 \cdot 3^{n-1}$$ 5. **Compare with given options:** - A: $3 \cdot 5^n$ (incorrect, bases and powers do not match) - B: $3 \cdot 5^{n-1}$ (incorrect, base should be 3, not 5) - C: $5 \cdot 3^n$ (incorrect, exponent should be $n-1$) - D: $5 \cdot 3^{n-1}$ (correct) **Final answer:** $$\boxed{f(n) = 5 \cdot 3^{n-1}}$$