To create an algebraic expression for the number of pieces in figure $n$, let's analyze the pattern:

• Figure 1: 1 piece
• Figure 2: 3 pieces
• Figure 3: 6 pieces
• Figure 4: 10 pieces

We need to find a relationship between the figure number $n$ and the number of pieces.

Step 1: Identify the Pattern

Let's list the number of pieces:

• $1, 3, 6, 10, \dots$

These numbers look like triangular numbers, which are generated by summing the first $n$ natural numbers:

• $T_1 = 1$
• $T_2 = 1 + 2 = 3$
• $T_3 = 1 + 2 + 3 = 6$
• $T_4 = 1 + 2 + 3 + 4 = 10$

So, the number of pieces in figure $n$ corresponds to the $n$-th triangular number.

Step 2: Formula for Triangular Numbers

The formula for the $n$-th triangular number $T_n$ is given by:

$T_n = \frac{n(n + 1)}{2}$

This formula represents the sum of the first $n$ natural numbers.

Step 3: General Formula for Number of Pieces

Thus, the number of pieces in figure $n$ is:

$P(n) = \frac{n(n + 1)}{2}$

Where:

• $P(n)$ represents the number of pieces in figure $n$,
• $n$ is the figure number.

Step 4: Verification

Let’s check this formula for the given figures:

• Figure 1: $P(1) = \frac{1(1 + 1)}{2} = \frac{1 \times 2}{2} = 1$
• Figure 2: $P(2) = \frac{2(2 + 1)}{2} = \frac{2 \times 3}{2} = 3$
• Figure 3: $P(3) = \frac{3(3 + 1)}{2} = \frac{3 \times 4}{2} = 6$
• Figure 4: $P(4) = \frac{4(4 + 1)}{2} = \frac{4 \times 5}{2} = 10$

The formula works correctly for the provided figures.

Final Formula:

$P(n) = \frac{n(n + 1)}{2}$

This is the algebraic expression for the number of pieces in figure $n$.

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Would you like more details on how triangular numbers work, or do you have any specific questions?

5 Related Questions:

1. How do you derive the formula for the sum of the first $n$ natural numbers?
2. Can this formula be used for non-integer values of $n$?
3. What are some real-world applications of triangular numbers?
4. How does the growth rate of triangular numbers compare to square or cubic numbers?
5. Can we generalize this to other patterns of shapes besides triangles?

Tip:

Always verify your formula by checking it with specific values to ensure accuracy.