To find the recursive formula for the sequence $a_n$ based on the explicit formula $a_n = -19n - 2$, we need to express $a_n$ in terms of $a_{n-1}$.

Step 1: Start with the explicit formula

$a_n = -19n - 2$

Step 2: Express $a_{n-1}$ using the explicit formula

Substitute $n-1$ into the formula to get the value for $a_{n-1}$: $a_{n-1} = -19(n-1) - 2$ Now simplify this: $a_{n-1} = -19n + 19 - 2$ $a_{n-1} = -19n + 17$

Step 3: Relate $a_n$ and $a_{n-1}$

Now, we want to express $a_n$ using $a_{n-1}$. We know: $a_n = -19n - 2$ and we found: $a_{n-1} = -19n + 17$ We can solve for $a_n$ in terms of $a_{n-1}$ by noting that: $a_n = a_{n-1} - 19$

Final Recursive Formula:

$a_n = a_{n-1} - 19$ This is the recursive formula for the sequence, and it depends on the previous term $a_{n-1}$.

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Would you like more details or have any questions about the process?

Here are 5 related questions you can explore:

1. How do we derive the general form of a recursive formula from an explicit formula?
2. What is the significance of a recursive formula in sequences?
3. Can recursive formulas be used for more complex sequences, like geometric or exponential sequences?
4. How can we convert a recursive sequence back into an explicit formula?
5. What is the base case needed for a recursive formula?

Tip: When working with sequences, always identify the relationship between consecutive terms to derive recursive formulas.