Let's work through the arithmetic sequence:

The sequence is:
$8, -5, -18, -31, \dots$

Step 1: Find the common difference (d)

To find the common difference, subtract any term from the next term in the sequence:

$d = -5 - 8 = -13$ $d = -18 - (-5) = -13$ $d = -31 - (-18) = -13$

So, the common difference $d$ is $-13$.

Step 2: Define the recursive formula

For an arithmetic sequence, the recursive formula is:

$d(1) = \text{the first term of the sequence}$ $d(n) = d(n-1) + d$

Given the first term is 8 and the common difference is $-13$, we can write:

$d(1) = 8$ $d(n) = d(n-1) - 13 \quad \text{for} \quad n > 1$

Final Answer:

• $d(1) = 8$
• $d(n) = d(n-1) - 13$ for $n > 1$

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Would you like more details on this solution or have any questions?

Here are some related questions:

1. How would you write the explicit formula for this arithmetic sequence?
2. How do you find the nth term of an arithmetic sequence using the explicit formula?
3. Can you apply the recursive formula to find the 5th term of this sequence?
4. How does the common difference affect the behavior of an arithmetic sequence?
5. How can the concept of arithmetic sequences be applied to real-life situations?

Tip: The recursive formula is especially useful for finding terms in a sequence when you know the previous term, but the explicit formula is faster for directly finding any term $n$.