https://file.aiquickdraw.com/mathbot/file/1728573674102sptoafpx.jpg

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**.

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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.

The graph shows a sequence. What are the explicit and recursive forms of the sequence?

Learn how to find the explicit and recursive forms of an arithmetic sequence shown on a graph. This problem involves identifying the common difference and applying the correct formulas for both forms. Suitable for Grades 8-10.

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