Question

1. here is a visual pattern.

problems 5 - 6. this table shows the number of tiles in figures 1 - 3.

1. do you agree that figure 5 will have 24 + 6 tiles? circle one. explain your reasoning.
2. how many tiles will there be in figure 6?

Explanation:

Step1: Identify the pattern

The number of tiles in each figure forms an arithmetic - sequence. Figure 1 has 6 tiles, Figure 2 has 12 tiles, Figure 3 has 18 tiles. The common difference is 6, and the formula for the number of tiles in the \(n\)th figure is \(a_n=6n\).

Step2: Describe Figure 4

Since the pattern seems to be that the figures are rectangular arrangements of tiles and the number of tiles increases by 6 each time. Figure 4 will be a rectangular arrangement of tiles. It will have 4 rows and 3 columns (or 3 rows and 4 columns) of tiles.

Step3: Calculate number of tiles in Figure 4

Using the formula \(a_n = 6n\), when \(n = 4\), \(a_4=6\times4 = 24\).

Step4: Analyze Figure 5

For Figure 5, using the formula \(a_n=6n\), when \(n = 5\), \(a_5=6\times5=30\). And \(24 + 6=30\), so Figure 5 will have \(24 + 6\) tiles. The answer to question 5 is Yes.

Step5: Calculate number of tiles in Figure 6

Using the formula \(a_n = 6n\), when \(n = 6\), \(a_6=6\times6=36\).

Answer:

a. Figure 4 will be a rectangular arrangement of 24 tiles, with 4 rows and 6 columns (or 6 rows and 4 columns).
b. 24

1. Yes. The number of tiles in the \(n\)th figure follows the formula \(a_n = 6n\). For \(n = 5\), \(a_5=6\times5 = 30\), and \(24+6 = 30\).
2. 36