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The problem provides a pattern of octagons, and you are asked to write two equivalent expressions that represent the number of line segments in figure \( n \).

### Step-by-step Analysis:

1. **Figure 1:** There is one octagon. An octagon has 8 sides (or line segments).
2. **Figure 2:** There are two octagons. Each octagon has 8 sides, and the octagons share one side where they connect. So, the total number of line segments is \( 8 + 8 - 1 = 15 \).
3. **Figure 3:** There are three octagons. The first two octagons share one side, and the second and third octagons share one side. So, the total number of line segments is \( 8 + 8 - 1 + 8 - 1 = 22 \).

In general, for figure \( n \) with \( n \) octagons:
- Each octagon contributes 8 sides.
- There are \( n-1 \) shared sides (because each pair of connected octagons shares a side).

### Expressions for Figure \( n \):
1. **Expression 1:**
   Total number of sides = \( 8n \) (number of octagons times 8 sides per octagon).
   
   Total number of shared sides = \( n-1 \).
   
   So, the total number of line segments is:
   \[
   8n - (n-1) = 8n - n + 1 = 7n + 1
   \]

2. **Expression 2:** 
   An alternative way to think about it is by adding 7 sides for each additional octagon after the first one (since they share one side):
   \[
   8 + 7(n-1) = 8 + 7n - 7 = 7n + 1
   \]

Thus, the two equivalent expressions are:
\[
7n + 1 \quad \text{and} \quad 8 + 7(n-1)
\]

Would you like further details or have any questions?

### Related Questions:
1. How many line segments are there in figure 4 based on the pattern?
2. What is the total number of line segments if the number of octagons is 10?
3. How does the number of shared sides change as \( n \) increases?
4. What would the expression look like if each polygon had 6 sides instead of 8?
5. Can you write a recursive formula to describe the number of sides for each figure?

**Tip:** Look for repeating patterns and how elements are shared between shapes to simplify complex counting problems.

Examine the pattern of figures. Write two equivalent expressions that can each be used to represent the number of line segments in figure n.

Learn how to find equivalent expressions that represent the number of line segments in a sequence of geometric figures. This problem covers pattern recognition and algebraic expressions suitable for students in Grades 6-8.

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