The sequence given in the image is:

$5, -4, -13, -22, \dots$

1. Explicit Formula (Arithmetic Sequence):

This appears to be an arithmetic sequence, where each term is obtained by subtracting a constant from the previous one. Let’s calculate the common difference:

$-4 - 5 = -9$ $-13 - (-4) = -9$ $-22 - (-13) = -9$

The common difference $d = -9$.

The explicit formula for an arithmetic sequence is:

$a_n = a_1 + (n-1) \cdot d$

Where:

• $a_n$ is the $n$-th term,
• $a_1 = 5$ is the first term,
• $d = -9$ is the common difference.

Thus, the explicit formula is:

$a_n = 5 + (n-1)(-9)$ Simplifying:

$a_n = 5 - 9(n-1)$ $a_n = 5 - 9n + 9$ $a_n = 14 - 9n$

So, the explicit formula is:

$a_n = 14 - 9n$

2. Recursive Formula:

For the recursive formula of an arithmetic sequence, we use the form:

$a_1 = 5$ $a_n = a_{n-1} - 9 \quad \text{for} \quad n \geq 2$

This shows that each term is obtained by subtracting 9 from the previous term.

Summary:

• Explicit: $a_n = 14 - 9n$
• Recursive:
  • $a_1 = 5$
  • $a_n = a_{n-1} - 9$, for $n \geq 2$

Would you like any further clarifications or more details?

Here are some related questions you might consider:

1. What is the 10th term of the sequence using the explicit formula?
2. How would the sequence change if the common difference was positive?
3. Can an arithmetic sequence have a common difference of 0?
4. How do explicit and recursive formulas compare for other types of sequences, like geometric sequences?
5. What is the sum of the first 10 terms of this sequence?

Tip: The explicit formula is more useful when you want to calculate a distant term in the sequence, while the recursive formula is better for building the sequence term by term.