To define the given geometric sequence $1, -\frac{1}{5}, \frac{1}{25}, -\frac{1}{125}, \dots$ recursively:

1. The first term ($a_1$) is given as $1$.
2. Each subsequent term ($a_n$ for $n \geq 2$) is obtained by multiplying the previous term ($a_{n-1}$) by the common ratio. The common ratio in this sequence is $-\frac{1}{5}$.

Thus, the recursive formula is:

$a_1 = 1$ $a_n = a_{n-1} \times \left(-\frac{1}{5}\right), \quad n \geq 2$

Explanation:

• The first term $a_1$ is $1$.
• For $n \geq 2$, multiply the previous term $a_{n-1}$ by $-\frac{1}{5}$ to get the next term. For example:
  • $a_2 = 1 \times \left(-\frac{1}{5}\right) = -\frac{1}{5}$,
  • $a_3 = -\frac{1}{5} \times \left(-\frac{1}{5}\right) = \frac{1}{25}$,
  • $a_4 = \frac{1}{25} \times \left(-\frac{1}{5}\right) = -\frac{1}{125}$.

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Would you like a detailed step-by-step derivation of this formula? Here are 5 related questions for further exploration:

1. How can you write an explicit formula for this geometric sequence?
2. What is the sum of the first $n$ terms of this sequence?
3. How can you determine the $n$-th term of a geometric sequence given $a_1$ and the common ratio?
4. What happens if the common ratio is positive? How does the sequence change?
5. How does the recursive formula differ from an explicit formula?

Tip: In a recursive formula, always start by clearly specifying the initial term(s).