A geometric sequence follows the formula:

$a_n = a_1 \cdot r^{(n-1)}$

where:

• $a_n$ is the $n$th term,
• $a_1 = 3$ (given),
• $n = 8$,
• $a_8 = 384$,
• $r$ is the common ratio.

Step 1: Substitute the known values into the formula

$384 = 3 \cdot r^{(8-1)}$

$384 = 3 \cdot r^7$

Step 2: Solve for $r$

First, divide both sides by 3:

$\frac{384}{3} = r^7$

$128 = r^7$

Now, take the seventh root of both sides:

$r = \sqrt[7]{128}$

Since $128 = 2^7$, we get:

$r = 2$

Final Answer:

The common ratio is 2.

Would you like a step-by-step explanation for a different approach or more details?

Here are some related questions for further understanding:

1. How do you find the sum of a geometric sequence?
2. What happens if the common ratio is negative?
3. How do you determine the common ratio if given two non-consecutive terms?
4. How does the common ratio affect the growth of a geometric sequence?
5. How can geometric sequences be applied in real-life situations?

Tip: In a geometric sequence, the common ratio remains constant between consecutive terms. Always check by dividing one term by the previous term.