The given sequence is $8, 24, 72, 216$.

We want to determine which explicit definition matches this sequence. Let's analyze:

Step 1: Examine the sequence

• First term ($a_1$): $8$
• Second term ($a_2$): $24$
• Third term ($a_3$): $72$
• Fourth term ($a_4$): $216$

Step 2: Check for a pattern

Each term increases by a multiplication factor: $\frac{24}{8} = 3, \quad \frac{72}{24} = 3, \quad \frac{216}{72} = 3$ The terms are multiplied by $3$ successively, so this is a geometric sequence.

Step 3: General formula for a geometric sequence

The formula for a geometric sequence is: $a_n = a_1 \cdot r^{n-1}$ where:

• $a_1 = 8$ (the first term),
• $r = 3$ (the common ratio).

Substituting these values: $a_n = 8 \cdot 3^{n-1}$

Step 4: Match with the provided options

Looking at the choices:

• $a_n = 3^{n-1} \cdot 8$: This is correct and matches our formula.

The correct answer is: $\boxed{a_n = 3^{n-1} \cdot 8}$

Would you like a detailed breakdown or further examples of geometric sequences?

Related Questions:

1. What is the explicit formula for a sequence where the common ratio is $2$?
2. How do you derive the formula for a geometric sequence?
3. Can geometric sequences have a fractional common ratio?
4. What happens to a geometric sequence as $n \to \infty$ if $0 < r < 1$?
5. How can arithmetic and geometric sequences be distinguished?

Tip:

When working with sequences, always determine whether the pattern is based on addition (arithmetic) or multiplication (geometric). This helps you select the right formula.