https://file.aiquickdraw.com/mathbot/file/1727113892566jx4y8gqj.jpg

In the image, you have a question related to sequence representations that involves plotting points on a graph, and you are required to find both the recursive equation and the explicit equation of the sequence.

### Step-by-step approach:

#### 1. **Analyzing the Points**
The points on the graph appear to be at:
- \( (0, 5) \)
- \( (1, 3) \)
- \( (2, 1) \)
- \( (3, -1) \)

These points suggest that the sequence is linear, as the differences between successive points on the y-axis are constant.

#### 2. **Finding the Recursive Equation**
A recursive formula expresses each term in the sequence in relation to the previous term.

- The first term \( a_1 \) is the value of the sequence at \( x = 0 \), which is \( a_1 = 5 \).
- The common difference (the difference between successive terms) is \( 3 - 5 = -2 \).

So, the recursive equation will be:
\[
a_n = a_{n-1} - 2 \quad \text{with} \quad a_1 = 5
\]

#### 3. **Finding the Explicit Equation**
To write the explicit equation, we use the form of the equation for an arithmetic sequence:
\[
a_n = a_1 + (n-1) \cdot d
\]
where \( a_1 \) is the first term and \( d \) is the common difference.

Substituting \( a_1 = 5 \) and \( d = -2 \):
\[
a_n = 5 + (n-1) \cdot (-2)
\]
Simplifying:
\[
a_n = 5 - 2(n-1)
\]
\[
a_n = 5 - 2n + 2
\]
\[
a_n = 7 - 2n
\]

Thus, the explicit equation is:
\[
a_n = 7 - 2n
\]

### Conclusion:
- **Recursive equation:** \( a_n = a_{n-1} - 2 \) with \( a_1 = 5 \)
- **Explicit equation:** \( a_n = 7 - 2n \)

Would you like further clarification on any step?

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Here are some related questions to explore further:
1. What is the significance of a recursive equation in sequences?
2. How can you determine if a sequence is arithmetic from a graph?
3. Can a sequence have both recursive and explicit representations?
4. What are some real-life applications of arithmetic sequences?
5. How would the equations change if the common difference were positive?

**Tip:** When analyzing sequences, always check if the differences between terms are consistent to identify arithmetic sequences.

Given the sequence representation below, complete the recursive and explicit equations.

Learn how to find both the recursive and explicit formulas for an arithmetic sequence from a graphical representation. Includes step-by-step solutions and explanations. Suitable for students in Grades 8-10.

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