Problem Analysis:
The image shows a sequence of three triangles, each divided into smaller triangles. The pattern involves blue and yellow triangles, and the task is to identify the underlying rule or pattern governing the sequence.
#### Observations:
1.
First Triangle:
- The large triangle is divided into 4 smaller triangles.
- One central triangle is blue, and the other three are yellow.
2.
Second Triangle:
- Each of the 4 smaller triangles from the first step is further divided into 4 even smaller triangles.
- This results in a total of \(4 \times 4 = 16\) smaller triangles.
- The central triangle of each of the 4 original smaller triangles is blue, and the rest are yellow.
- This means there are \(4\) blue triangles (one in each of the 4 regions).
3.
Third Triangle:
- Each of the 16 smaller triangles from the second step is further divided into 4 even smaller triangles.
- This results in a total of \(16 \times 4 = 64\) smaller triangles.
- The central triangle of each of the 16 original smaller triangles is blue, and the rest are yellow.
- This means there are \(16\) blue triangles (one in each of the 16 regions).
Pattern Identification:
- In the
first triangle, there is \(1\) blue triangle.
- In the
second triangle, there are \(4\) blue triangles.
- In the
third triangle, there are \(16\) blue triangles.
The number of blue triangles follows a pattern:
\[
1, 4, 16, \dots
\]
This is a geometric sequence where each term is multiplied by \(4\) to get the next term:
\[
1 \times 4 = 4, \quad 4 \times 4 = 16, \quad \text{and so on.}
\]
General Rule:
If we denote the number of blue triangles in the \(n\)-th step as \(B_n\), then:
\[
B_n = 4^{n-1}
\]
where \(n\) is the step number.
Verification:
- For \(n = 1\): \(B_1 = 4^{1-1} = 4^0 = 1\)
- For \(n = 2\): \(B_2 = 4^{2-1} = 4^1 = 4\)
- For \(n = 3\): \(B_3 = 4^{3-1} = 4^2 = 16\)
This matches the observed pattern.
Conclusion:
The number of blue triangles in the \(n\)-th step is given by the formula:
\[
\boxed{4^{n-1}}
\]
Parent Tip: Review the logic above to help your child master the concept of math patterns.
