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To predict the perimeter of the 20th figure in this sequence, let's start by analyzing the pattern.

We are given figures that increase in size, where each figure consists of squares arranged in a triangular pattern. Here's the approach:

1. **Observation of the first few figures:**
   - Figure 1 has 1 square, arranged as a 1x1 block.
   - Figure 2 has 3 squares, arranged in a 2x2 triangular shape.
   - Figure 3 has 6 squares, arranged in a 3x3 triangular shape, and so on.

The shape grows in both height and width, forming a larger triangular structure.

2. **Perimeter Formula:**
   - For these triangular figures, the height and width of figure \(n\) are both \(n\). The perimeter \(P_n\) of figure \(n\) seems to follow a linear relationship based on the squares added.
   - Observing the perimeter for the first few figures, we can calculate:
     - Figure 1 has a perimeter of 4 units.
     - Figure 2 has a perimeter of 6 units.
     - Figure 3 has a perimeter of 8 units.
     - Figure 4 has a perimeter of 10 units.
     - Figure 5 has a perimeter of 12 units.

   It appears that the perimeter increases by 2 units for each subsequent figure.

3. **Prediction for Figure 20:**
   Since the perimeter increases by 2 units with each figure, we can establish the relationship:
   \[
   P_n = 2n + 2
   \]
   For figure 20:
   \[
   P_{20} = 2(20) + 2 = 42 \, \text{units}
   \]

Thus, the perimeter of the 20th figure is **42 units**.

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Would you like more details on this approach or further clarification?

Here are 5 related questions:
1. How is the number of squares related to the figure number?
2. How does the area of the figures grow as \(n\) increases?
3. Can we derive a general formula for the perimeter of any figure \(n\)?
4. How would the perimeter change if each square was 2x2 units instead of 1x1?
5. What other geometric shapes follow a similar growth pattern?

**Tip:** Identifying patterns in sequences early on helps in predicting future values accurately.

Predict the perimeter of the 20th figure. Each figure is made of squares that are 1 unit by 1 unit.

Learn how to predict the perimeter of a geometric figure based on a repeating pattern. Using the formula P(n) = 2n + 2, we calculate the perimeter for the 20th figure, arriving at a result of 42 units. Suitable for students in Grades 5-7.

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