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.