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.