https://file.aiquickdraw.com/mathbot/file/17285141255860wluu4rx.jpg

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.

Use the initial term and the recursive formula to find an explicit formula for the sequence a_n. The initial term is a_1 = -19, and the recursive formula is a_n = a_{n-1} - 10.

Learn how to derive the explicit formula for an arithmetic sequence given the recursive formula a_n = a_{n-1} - 10 and the initial term a_1 = -19. The problem involves arithmetic sequences and recursive relations, suitable for students in Grades 9-11.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation