https://file.aiquickdraw.com/mathbot/file/1726423485623g2cmntxi.jpg

The image shows a series of house-like figures, each consisting of a triangle on top of a square. We are tasked with finding the generalized formula for the number of sides in these shapes as the sequence progresses.

### Step-by-Step Analysis:

1. **First figure**: The house has 5 sides (3 for the triangle and 4 for the square, but two sides are shared).
2. **Second figure**: Another house is added next to the first, but they share a common side, resulting in 9 sides (5 from the first house and 4 from the second, minus the shared side).
3. **Third figure**: Again, a house is added sharing a side with the previous one, giving a total of 13 sides.
4. **Fourth figure**: Following the same pattern, the fourth figure has 17 sides.

### Generalizing the Pattern:
Each house added introduces 4 new sides (5 sides per house minus 1 shared side). So, the formula for the number of sides \( S_n \) in the sequence, where \( n \) is the number of houses, is:
\[
S_n = 5 + 4(n - 1) = 4n + 1
\]

### Finding the 7th and 8th Terms:
Using the formula \( S_n = 4n + 1 \):
- For the 7th term:
\[
S_7 = 4(7) + 1 = 28 + 1 = 29
\]
- For the 8th term:
\[
S_8 = 4(8) + 1 = 32 + 1 = 33
\]

Thus, the 7th term has 29 sides, and the 8th term has 33 sides.

### Do you want more details or have any questions?

### Related Questions:
1. How can you derive a formula if the shape pattern changes?
2. What is the 10th term in this sequence?
3. How does this pattern relate to sequences in arithmetic progressions?
4. If two shapes are added at a time, what would the new formula be?
5. What happens to the sequence if the shapes don't share any sides?

### Tip:
Always check for a pattern in how figures grow or overlap to find a general formula.

Write the generalized statement for the number of sides and find the 7th and 8th term.

Explore the geometric sequence of houses made of triangles and squares. Learn how to generalize the number of sides in each step and find the 7th and 8th terms of the pattern using the formula S_n = 4n + 1. This problem is ideal for students in Grades 6-8.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation