https://file.aiquickdraw.com/mathbot/file/1726877449011f4kd4dt4.jpg

Let's go over the questions step by step based on the image.

1. **Describe how the rectangle grows from Step 1 to Step 3**:
   - In Step 1, you can observe that there is a single square.
   - In Step 2, the rectangle has grown to a 2x2 grid (4 squares total).
   - In Step 3, the grid expands further to a 3x3 grid (9 squares total).
   
   The growth pattern appears to be a square pattern, where each step increases both the width and height by 1 unit, forming larger squares.

2. **Write an equation for sequence \( S \)**:
   - The number of squares follows a pattern that matches perfect squares. In Step 1, there is \( 1^2 = 1 \) square, in Step 2, there are \( 2^2 = 4 \) squares, and in Step 3, there are \( 3^2 = 9 \) squares.
   - Therefore, for step \( n \), the number of squares \( S(n) \) is given by:
     \[
     S(n) = n^2
     \]

3. **Is \( S \) a geometric sequence, an arithmetic sequence, or neither?**:
   - This sequence is **neither arithmetic nor geometric**.
   - An arithmetic sequence has a constant difference between consecutive terms, and a geometric sequence has a constant ratio between terms. Here, the differences between terms (1, 4, 9, etc.) are not constant, and neither are the ratios.
   - Instead, this is a quadratic sequence, as the number of squares is the square of the step number.

Would you like further explanation on any part of the solution?

Here are five related questions to expand on this:
1. What is the difference between an arithmetic and geometric sequence?
2. How can you determine if a sequence follows a quadratic pattern?
3. How would the equation change if the rectangles grew by adding only one row or column each step instead of both?
4. How could you graph this sequence of squares versus the step number?
5. What is the relationship between the area of the growing rectangles and the step number?

**Tip**: In sequences involving geometric shapes, checking for patterns in areas or volumes can often lead to quadratic or cubic expressions.

Describe how the rectangle grows from Step 1 to Step 3. Write an equation for sequence S, so that S(n) is the number of squares in Step n. Is S a geometric sequence, an arithmetic sequence, or neither? Explain how you know.

Analyze a growing rectangle pattern and learn how to describe its growth using the quadratic sequence formula S(n) = n^2. Understand why this sequence is neither arithmetic nor geometric. Suitable for students in Grades 7-9.

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