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?

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.