To determine the recursive formula for the sequence $-1, -5, -25, -125$, observe the pattern of multiplication between consecutive terms:

Steps:

1. The first term is $a_1 = -1$.
2. Each term is obtained by multiplying the previous term by $-5$.

Recursive Formula:

$a_n = a_{n-1} \cdot (-5), \quad \text{for } n \geq 2$

Complete Specification:

$a_1 = -1, \quad a_n = a_{n-1} \cdot (-5) \text{ for } n \geq 2$

This defines the sequence recursively.

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Do you want further details or have any questions? Here are five related questions:

1. How do you write an explicit formula for this sequence?
2. What is the $n$-th term of this sequence explicitly?
3. How do you derive recursive formulas for more complex sequences?
4. Can this sequence be represented as a geometric progression?
5. What is the sum of the first $n$ terms of this sequence?

Tip: Always check the ratio or difference between terms to identify whether a sequence is arithmetic, geometric, or neither!