In a geometric sequence, the general term $g_n$ is given by the formula:

$g_n = g_1 \cdot r^{n-1}$

Here:

• $g_2 = 24$ (second term),
• $r = \frac{1}{2}$ (common ratio).

Step 1: Find $g_1$

Using the formula for the second term:

$g_2 = g_1 \cdot r^{2-1}$

Substitute $g_2 = 24$ and $r = \frac{1}{2}$:

$24 = g_1 \cdot \frac{1}{2}$

Multiply both sides by 2:

$g_1 = 48$

Step 2: Write the general expression for $g_n$

Now that we know $g_1 = 48$, substitute it into the general formula:

$g_n = 48 \cdot \left(\frac{1}{2}\right)^{n-1}$

Step 3: Find $g_5$

Substitute $n = 5$ into the formula:

$g_5 = 48 \cdot \left(\frac{1}{2}\right)^{5-1}$

$g_5 = 48 \cdot \left(\frac{1}{2}\right)^4$

$g_5 = 48 \cdot \frac{1}{16}$

$g_5 = 3$

Final Results:

1. General expression for $g_n$: $g_n = 48 \cdot \left(\frac{1}{2}\right)^{n-1}$

2. Value of $g_5$: $g_5 = 3$

Would you like further clarification or details?

---

Here are 5 related questions to consider:

1. What is the sum of the first $n$ terms of a geometric sequence?
2. How does the common ratio affect the behavior of a geometric sequence?
3. Can you derive the nth term for a geometric sequence with a negative common ratio?
4. What happens to $g_n$ as $n \to \infty$ in this sequence?
5. How do you determine if a sequence is geometric based on a given set of terms?

Tip: Always double-check your calculations, especially when working with exponents and fractions!