The problem provides a recursive formula and an initial term for the sequence $a_n$. The given information is:

• Initial term: $a_1 = -19$
• Recursive formula: $a_n = a_{n-1} - 10$

Task:

We need to find an explicit formula for $a_n$, the general term of the sequence.

Solution:

1. The recursive relation $a_n = a_{n-1} - 10$ suggests that each term is decreasing by 10 from the previous term.
2. We recognize this as an arithmetic sequence where the common difference $d = -10$.
3. The general explicit formula for an arithmetic sequence is: $a_n = a_1 + (n - 1) \cdot d$ Substituting the values for $a_1 = -19$ and $d = -10$: $a_n = -19 + (n - 1) \cdot (-10)$
4. Simplify the expression: $a_n = -19 - 10(n - 1)$ $a_n = -19 - 10n + 10$ $a_n = -10n - 9$

Final explicit formula:

$a_n = -10n - 9$

Let me know if you'd like further explanation or additional examples.

Follow-up questions:

1. How would the formula change if the common difference was positive?
2. Can you derive the explicit formula if $a_1$ was a different value?
3. What happens to the sequence as $n \to \infty$?
4. How can you verify the explicit formula using the recursive one?
5. What is the 5th term of the sequence using the explicit formula?

Tip:

For arithmetic sequences, always remember that the difference between consecutive terms is constant. The explicit formula can be found by relating this difference to the initial term.