figure 1 has 12 pieces figure 2 has 23 pieces figure 3  has 38 pieces so according to this order make formulaa

To derive a formula based on the number of pieces in the figures, we first need to analyze the pattern. The given sequence is:

- Figure 1 has 12 pieces.
- Figure 2 has 23 pieces.
- Figure 3 has 38 pieces.

Let's denote the number of pieces in figure \( n \) as \( P(n) \).

We can observe that the number of pieces increases as \( n \) increases, and we aim to find a relationship between \( n \) and \( P(n) \).

### Step 1: Check Differences
We first calculate the differences between consecutive terms:
- \( P(2) - P(1) = 23 - 12 = 11 \)
- \( P(3) - P(2) = 38 - 23 = 15 \)

The first differences are 11 and 15, showing that the change is not constant. Therefore, it's likely a quadratic pattern.

### Step 2: Second Differences
We calculate the second differences:
- \( 15 - 11 = 4 \)

Since the second differences are constant (4), this confirms that the pattern follows a **quadratic equation**.

### Step 3: General Form of Quadratic Equation
The general form of a quadratic equation is:
\[
P(n) = an^2 + bn + c
\]
We now need to find the coefficients \( a \), \( b \), and \( c \).

### Step 4: Set Up a System of Equations
Using the given points \( (1, 12) \), \( (2, 23) \), and \( (3, 38) \), we substitute into the quadratic equation:

1. For \( n = 1 \): \( a(1)^2 + b(1) + c = 12 \) ⟹ \( a + b + c = 12 \)
2. For \( n = 2 \): \( a(2)^2 + b(2) + c = 23 \) ⟹ \( 4a + 2b + c = 23 \)
3. For \( n = 3 \): \( a(3)^2 + b(3) + c = 38 \) ⟹ \( 9a + 3b + c = 38 \)

### Step 5: Solve the System of Equations
We now solve the system of three equations:

1. \( a + b + c = 12 \)
2. \( 4a + 2b + c = 23 \)
3. \( 9a + 3b + c = 38 \)

By solving this system, we find:
- \( a = 2 \)
- \( b = 5 \)
- \( c = 5 \)

### Step 6: Final Formula
The formula for the number of pieces in figure \( n \) is:
\[
P(n) = 2n^2 + 5n + 5
\]

Would you like more details on solving the system of equations?

### Related Questions:
1. How do we verify the correctness of the formula with additional figures?
2. What is the importance of second differences in identifying quadratic patterns?
3. Can we generalize this approach to other types of patterns (linear, cubic)?
4. How does solving systems of equations help in finding unknown coefficients?
5. What are the potential applications of quadratic sequences in real-life situations?

### Tip:
When faced with number patterns, always check first and second differences to identify if the sequence is linear, quadratic, or higher degree.

figure 1 has 12 pieces figure 2 has 23 pieces figure 3 has 38 pieces so according to this order make formulaa

Explore how to derive a quadratic formula based on a sequence of figures with 12, 23, and 38 pieces. This detailed step-by-step solution covers algebraic concepts like quadratic equations and solving systems of equations. Suitable for students in Grades 9-12.

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