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The image shows a growing pattern of rectangles, and the problem asks to write an equation for the sequence \( S(n) \), where \( S(n) \) represents the number of squares in Step \( n \) of the pattern.

Let's analyze the pattern:

- The first rectangle has 2 squares.
- The second rectangle has 6 squares.
- The third rectangle has 12 squares.
- The fourth rectangle has 20 squares.
- The fifth rectangle has 30 squares.

This appears to follow a quadratic pattern, as the difference between the number of squares in consecutive steps increases by 2 each time:
- Difference between Step 1 and Step 2: \( 6 - 2 = 4 \)
- Difference between Step 2 and Step 3: \( 12 - 6 = 6 \)
- Difference between Step 3 and Step 4: \( 20 - 12 = 8 \)
- Difference between Step 4 and Step 5: \( 30 - 20 = 10 \)

The second differences are constant, indicating that the sequence can be modeled by a quadratic equation of the form:
\[
S(n) = an^2 + bn + c
\]
To find the values of \( a \), \( b \), and \( c \), we can use the known values of \( S(n) \):

1. \( S(1) = 2 \)
2. \( S(2) = 6 \)
3. \( S(3) = 12 \)

We substitute these into the quadratic equation to form a system of equations:

1. \( a(1)^2 + b(1) + c = 2 \)  →  \( a + b + c = 2 \)
2. \( a(2)^2 + b(2) + c = 6 \)  →  \( 4a + 2b + c = 6 \)
3. \( a(3)^2 + b(3) + c = 12 \) →  \( 9a + 3b + c = 12 \)

We now solve this system of equations to find \( a \), \( b \), and \( c \).

Let me calculate that.The solution to the system of equations is:

- \( a = 1 \)
- \( b = 1 \)
- \( c = 0 \)

Thus, the equation for the sequence \( S(n) \), which gives the number of squares in Step \( n \), is:

\[
S(n) = n^2 + n
\]

This equation will generate the number of squares at each step in the pattern.

Would you like further clarification or additional details on how this equation works?

Here are 5 related questions to expand on this:

1. What does the term \( n^2 \) represent in the context of this pattern?
2. How can you derive the quadratic pattern from other visual growth sequences?
3. How do the first and second differences help identify a quadratic sequence?
4. Can this sequence be generalized for other similar rectangle growth patterns?
5. How would the equation change if the pattern started with 3 squares instead of 2?

**Tip:** Quadratic sequences have constant second differences, which helps you identify the form of the equation.

Write an equation for sequence S, so that S(n) is the number of squares in Step n of a growing pattern of rectangles.

Learn how to derive a quadratic equation for a growing rectangle pattern. Understand how to calculate the number of squares in Step n, using the quadratic sequence formula S(n) = n^2 + n. Suitable for Grades 7-9.

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