The graph provided represents an arithmetic sequence. The red points plot the sequence, and we need to determine the recursive formula based on the graph.

Steps:

1. Identify the first term $a_1$ from the graph.

   • From the graph, the first point (when $x = 0$) is at $y = 2$. Hence, the first term $a_1 = 2$.

2. Determine the common difference $d$ between consecutive terms.

   • From the graph, observe the difference in the $y$-values as $x$ increases.
   • When $x = 1$, $y = 4$, and when $x = 2$, $y = 6$, so the common difference $d = 4 - 2 = 2$.

Recursive Formula:

• The recursive formula for an arithmetic sequence is of the form: $a_n = a_{n-1} + d \quad \text{for} \, n \geq 2$ where $d$ is the common difference.
• From the graph, the common difference $d = 2$ and the first term $a_1 = 2$.

Thus, the recursive formula is: $a_1 = 2, \quad a_n = a_{n-1} + 2 \, \text{for} \, n \geq 2$

Would you like more details on this?

Here are five related questions:

1. How do you determine the explicit formula for an arithmetic sequence?
2. What happens to the recursive formula if the common difference is negative?
3. Can you describe the pattern in the sequence using the formula?
4. How does the graph of a geometric sequence differ from that of an arithmetic sequence?
5. What are the applications of arithmetic sequences in real-life problems?

Tip: In arithmetic sequences, the graph always forms a straight line, and the slope of that line equals the common difference.