https://file.aiquickdraw.com/mathbot/file/1731361912140kd6vuduc.jpg

To write a recursive rule for the sequence, let's analyze the sequence shown in the graph, where each point is given in the format \((n, a_n)\):

1. \( (1, 35) \)
2. \( (2, 24) \)
3. \( (3, 13) \)
4. \( (4, 2) \)

The sequence values are:
- \( a_1 = 35 \)
- \( a_2 = 24 \)
- \( a_3 = 13 \)
- \( a_4 = 2 \)

### Step 1: Determine the Pattern in the Sequence
To find a recursive rule, let's calculate the difference between consecutive terms:
- \( a_2 - a_1 = 24 - 35 = -11 \)
- \( a_3 - a_2 = 13 - 24 = -11 \)
- \( a_4 - a_3 = 2 - 13 = -11 \)

Each term in the sequence is obtained by subtracting 11 from the previous term.

### Step 2: Write the Recursive Rule
Since each term decreases by 11, we can express this with the following recursive formula:
\[
a_n = a_{n-1} - 11 \quad \text{for } n > 1
\]
with the initial condition:
\[
a_1 = 35
\]

### Final Answer
The recursive rule for the sequence is:
\[
a_1 = 35
\]
\[
a_n = a_{n-1} - 11 \quad \text{for } n > 1
\]

Would you like further details on this solution or any additional questions?

Here are some questions to expand on this topic:
1. How would the recursive rule change if the common difference were positive instead of negative?
2. Can we express this sequence using an explicit (non-recursive) formula?
3. How would you verify that a given sequence is arithmetic?
4. What are the applications of recursive sequences in real-world contexts?
5. How can we generalize this approach for different types of sequences, such as geometric sequences?

**Tip:** For any sequence, check the differences (or ratios) between terms to determine if it's arithmetic or geometric.

Write a recursive rule for the sequence based on the points provided in the graph.

Learn how to write a recursive rule for an arithmetic sequence using a set of points from a graph. This example demonstrates the use of recursive formulas in sequences, calculating differences between terms to identify a pattern, and establishing a rule for further terms. Suitable for students in Grades 8-10.

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