The given recursive formula for the geometric sequence is:

$a_n = 2a_{n-1}$

with an initial value of:

$a_1 = \frac{1}{8}$

Finding the Explicit Formula:

The general explicit formula for a geometric sequence is:

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

where:

• $a_1 = \frac{1}{8}$ (first term),
• $r = 2$ (common ratio).

Substituting these values:

$a_n = \frac{1}{8} \cdot 2^{(n-1)}$

Final Answer:

$a_n = \frac{1}{8} \cdot 2^{(n-1)}$

Would you like a step-by-step breakdown of how I derived this?

Related Questions:

1. How do you determine the common ratio in a geometric sequence?
2. What is the difference between recursive and explicit formulas?
3. How can you use the explicit formula to find $a_5$ in this sequence?
4. Can a geometric sequence have a negative common ratio? What happens in that case?
5. How would the formula change if the common ratio were $\frac{1}{2}$ instead of 2?

Tip:

To quickly verify your explicit formula, plug in small values of $n$ and see if they match the recursive sequence!