The graph in the image shows a sequence of points. Let's analyze the sequence step by step to determine both the explicit and recursive forms.

Step 1: Analyze the points

Looking at the graph, the points for the sequence correspond to the following values:

• When $x = 1$, $y = 5$
• When $x = 2$, $y = 7$
• When $x = 3$, $y = 9$
• When $x = 4$, $y = 11$
• When $x = 5$, $y = 13$

These values correspond to the terms of the sequence:

• $a_1 = 5$
• $a_2 = 7$
• $a_3 = 9$
• $a_4 = 11$
• $a_5 = 13$

Step 2: Find the pattern

We can observe that the sequence increases by a constant value of 2:

• $a_2 - a_1 = 7 - 5 = 2$
• $a_3 - a_2 = 9 - 7 = 2$
• $a_4 - a_3 = 11 - 9 = 2$
• $a_5 - a_4 = 13 - 11 = 2$

This means the sequence is arithmetic with a common difference of 2.

Step 3: Determine the explicit form

The explicit form of an arithmetic sequence is given by: $a_n = a_1 + (n - 1)d$ where $d$ is the common difference and $a_1$ is the first term.

Substituting $a_1 = 5$ and $d = 2$: $a_n = 5 + (n - 1)2 = 5 + 2n - 2 = 2n + 3$

So, the explicit form is $a_n = 2n + 3$.

Step 4: Determine the recursive form

The recursive form of an arithmetic sequence is: $a_1 = 5, \quad a_n = a_{n-1} + 2 \quad \text{for} \quad n > 1$

Step 5: Conclusion

The explicit and recursive forms of the sequence are:

• Explicit: $a_n = 2n + 3$
• Recursive: $a_1 = 5, \, a_n = a_{n-1} + 2 \, \text{for} \, n > 1$

Thus, the correct answer is Option B.

Would you like more details on any step or further explanations?

Here are 5 related questions to explore:

1. How can we generalize explicit and recursive forms for different types of sequences?
2. What is the formula for the $n$-th term of a geometric sequence?
3. How do you determine the common difference or ratio in a sequence?
4. Can a sequence be both arithmetic and geometric at the same time?
5. How do recursive sequences model real-world situations?

Tip: When identifying a sequence, always check if the differences between terms are constant for arithmetic or if the ratios between terms are constant for geometric sequences.