The given sequence is: $-5, -1, 3, 7, \dots$

To find an explicit formula for this sequence, we can observe that it is an arithmetic sequence because the difference between each consecutive term is constant.

Step 1: Find the common difference ($d$)

To find the common difference, subtract any term from the next term: $d = -1 - (-5) = 4$ The common difference is $d = 4$.

Step 2: Write the explicit formula

For an arithmetic sequence, the $n$-th term can be written as: $a_n = a_1 + (n - 1) \cdot d$ where $a_1$ is the first term and $d$ is the common difference.

In this case:

• $a_1 = -5$
• $d = 4$

So the explicit formula for the $n$-th term of the sequence is: $a_n = -5 + (n - 1) \cdot 4$ Simplify it: $a_n = -5 + 4n - 4$ $a_n = 4n - 9$

Thus, the explicit formula for the $n$-th term of the sequence is: $a_n = 4n - 9$

Let me know if you'd like any further explanation or have additional questions! Here are some related questions:

1. How would you find the sum of the first $n$ terms of this sequence?
2. What is the 10th term of this sequence?
3. How do you determine whether a sequence is arithmetic or geometric?
4. What would the sequence look like if the common difference were negative?
5. How would you derive a formula for a sequence that has a non-constant difference?

Tip: Always check if a sequence has a constant difference between consecutive terms to identify an arithmetic sequence!