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.

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).