Question

create a table, a graph, and an explicit equation to represent the number of squares in each figure.
figure 1
figure 2
figure 3

1. explicit equation:
2. graph:
3. table:

Explanation:

Step1: Count squares per figure

• Figure 1: 5 squares
• Figure 2: 13 squares
• Figure 3: 21 squares

Step2: Build the table

Define $n$ as figure number, $S(n)$ as number of squares.

$n$ (Figure Number)	1	2	3

Step3: Identify sequence pattern

Calculate the common difference:
$13-5=8$, $21-13=8$. This is an arithmetic sequence with first term $a_1=5$, common difference $d=8$.
The explicit formula for an arithmetic sequence is:
$$S(n) = a_1 + (n-1)d$$
Substitute values:
$$S(n) = 5 + (n-1) \times 8$$
Simplify:
$$S(n) = 8n - 3$$

Step4: Prepare graph points

Use the table values: $(1,5)$, $(2,13)$, $(3,21)$. Plot these on the grid, with the x-axis as figure number and y-axis as number of squares, then draw a straight line through the points.

Answer:

9. Table:

Figure Number	1	2	3

7. Explicit Equation:

$S(n) = 8n - 3$ (where $n$ is the figure number, $S(n)$ is the number of squares)

8. Graph:

Plot the points $(1, 5)$, $(2, 13)$, $(3, 21)$ on the grid, with the horizontal axis labeled "Figure Number" and vertical axis labeled "Number of Squares", then draw a straight line passing through these points.