Question

problem 6. figures 1, 2, and 3 are scaled copies of each other. each small square has a side - length of 1 unit and an area of 1 square unit. describe any patterns you see.

Explanation:

Step1: Count squares in Figure 1

There are 3 squares.

Step2: Count squares in Figure 2

There are 13 squares.

Step3: Count squares in Figure 3

There are 53 squares.

Step4: Analyze side - length relationship

Let's assume the side - length of the overall figure in Figure 1 is \(a_1\), Figure 2 is \(a_2\), and Figure 3 is \(a_3\). We can observe that the side - length of the non - L - shaped part seems to be increasing. Also, if we consider the number of squares \(n_1 = 3\), \(n_2=13\), \(n_3 = 53\). We can try to find a recurrence relation. Notice that \(n_2=4\times3 + 1=13\), \(n_3 = 4\times13+1 = 53\).

Answer:

The number of square units in each figure follows a pattern where each term (number of squares in a figure) is 4 times the previous term plus 1.