To write a recursive formula for the sequence $-3, -5, -7, -9, \dots$, we need to determine how each term relates to the previous term.

Step-by-step Derivation:

1. Identify the first term ($a_1$): The first term of the sequence is: $a_1 = -3$

2. Determine the common difference ($d$): The sequence decreases by 2 each time: $d = -5 - (-3) = -2$

3. Write the recursive formula: Each term is obtained by subtracting 2 from the previous term: $a_n = a_{n-1} - 2 \quad \text{for } n \geq 2$

Final Recursive Formula:

$a_1 = -3$ $a_n = a_{n-1} - 2, \quad \text{for } n \geq 2$

Let me know if you’d like further details or examples with this formula!

Related Questions:

1. How can the explicit formula for this sequence be derived?
2. What happens to the sequence if the common difference changes?
3. How can this recursive formula be implemented in a programming language like Python?
4. How does the concept of arithmetic sequences generalize to other series types?
5. Can you determine the $n$-th term directly without recursion?

Tip:

To check if your recursive formula is correct, compute a few terms manually and ensure they match the given sequence!