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The given explicit formula for the geometric sequence is:

\[
a_n = 2 \cdot 5^{(n-1)}
\]

In a recursive formula for a geometric sequence, each term is obtained by multiplying the previous term by a constant ratio (common ratio).

To express this recursively:
1. The first term \(a_1\) should match the explicit formula for \(n=1\). Substituting \(n=1\) into the explicit formula:

\[
a_1 = 2 \cdot 5^{(1-1)} = 2 \cdot 5^0 = 2
\]

So, the first term \(a_1\) is 2.

2. The recursive relationship must involve multiplying the previous term by the common ratio, which is 5. The recursive formula for the \(n\)-th term in a geometric sequence is:

\[
a_n = a_{n-1} \cdot r
\]

Here, the common ratio \(r = 5\).

Therefore, the correct recursive formula is:

\[
a_1 = 2, \quad a_n = a_{n-1} \cdot 5
\]

Looking at the options:

- **Option A**: \(a_1 = 2, \quad a_n = a_{n-1} \cdot 5\)

This matches the correct recursive form. Hence, the answer is **A**.

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Would you like further explanation on geometric sequences or any other details?

Here are 5 related questions to consider:
1. What is the difference between explicit and recursive formulas?
2. How can you derive a recursive formula from an explicit formula?
3. What is the significance of the common ratio in a geometric sequence?
4. Can the recursive formula apply to arithmetic sequences as well?
5. How do you convert a recursive sequence into an explicit form?

**Tip**: Always check if the recursive formula follows the pattern of multiplication (for geometric sequences) or addition (for arithmetic sequences).

Which of the following recursive formulas represents the same geometric sequence as the formula a_n = 2 * 5^(n-1)?

Explore how to represent a geometric sequence recursively. This problem features the sequence a_n = 2 * 5^(n-1) and helps students understand how to convert an explicit formula into a recursive one. Suitable for Grades 9-12.

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