Question 7

We will analyze the pattern of red squares and their area at each stage.

Step 1 — Counting red squares

Let's count the red squares at each stage.

At Stage 0, we have one large square.

$\text{Number of red squares at Stage 0} = \textbf{1}$

At Stage 1, the square is divided into 9 parts. The center part is removed. We are left with 8 red squares.

$\text{Number of red squares at Stage 1} = \textbf{8}$

At Stage 2, each of the 8 red squares is processed. Each red square becomes 8 smaller red squares.

$\text{Number of red squares at Stage 2} = 8 \times 8$

$= 8^2$

$\boxed{\textbf{64} \text{ red squares}}$

At Stage 3, each of the 64 red squares is processed. Each red square becomes 8 smaller red squares.

$\text{Number of red squares at Stage 3} = 64 \times 8$

$= 8^3$

$\boxed{\textbf{512} \text{ red squares}}$

Step 2 — Predicting future stages

We see a pattern for the number of red squares. It is $8^n$. Let's predict for Stage 4.

$\text{Number of red squares at Stage 4} = 8^4$

$= 512 \times 8$

$\boxed{\textbf{4096} \text{ red squares}}$

Now let's predict for Stage 5.

$\text{Number of red squares at Stage 5} = 8^5$

$= 4096 \times 8$

$\boxed{\textbf{32768} \text{ red squares}}$

Step 3 — Finding the rules for red squares

The number of red squares at stage $n$ follows a power of 8. The explicit formula directly gives the number for any stage $n$.

$\text{Explicit formula for } t_n = 8^n$

Here, $n$ is the stage number, starting from $n=0$. The recursive formula defines a term based on the previous term.

$\text{Recursive formula: } t_0 = \textbf{1}$

$\text{and } t_n = 8 \times t_{n-1} \text{ for } n \ge 1$

Step 4 — Calculating the area of the red region

Let the area of Stage 0 be 1 square unit. At each stage, the square is divided into 9 parts. One part is removed. So, $8/9$ of the area remains.

Area at Stage 0 is 1 square unit. Area at Stage 1 is $8/9$ of Stage 0 area.

$\text{Area at Stage 1} = 1 \times \frac{8}{9}$

$\boxed{\frac{\textbf{8}}{\textbf{9}} \text{ square units}}$

Area at Stage 2 is $8/9$ of Stage 1 area.

$\text{Area at Stage 2} = \frac{8}{9} \times \frac{8}{9}$

$= \left(\frac{8}{9}\right)^2$

$\boxed{\frac{\textbf{64}}{\textbf{81}} \text{ square units}}$

Area at Stage 3 is $8/9$ of Stage 2 area.

$\text{Area at Stage 3} = \left(\frac{8}{9}\right)^2 \times \frac{8}{9}$

$= \left(\frac{8}{9}\right)^3$

$\boxed{\frac{\textbf{512}}{\textbf{729}} \text{ square units}}$

Step 5 — Predicting future areas

Let's predict the area for Stage 4.

$\text{Area at Stage 4} = \left(\frac{8}{9}\right)^4$

$= \frac{4096}{6561}$

$\boxed{\frac{\textbf{4096}}{\textbf{6561}} \text{ square units}}$

Now let's predict the area for Stage 5.

$\text{Area at Stage 5} = \left(\frac{8}{9}\right)^5$

$= \frac{32768}{59049}$

$\boxed{\frac{\textbf{32768}}{\textbf{59049}} \text{ square units}}$

Step 6 — Finding the rules for area

The area at stage $n$ follows a power of $8/9$. The explicit formula directly gives the area for any stage $n$.

$\text{Explicit formula for } A_n = \left(\frac{8}{9}\right)^n$

Here, $n$ is the stage number, starting from $n=0$. The recursive formula defines an area based on the previous area.

$\text{Recursive formula: } A_0 = \textbf{1}$

$\text{and } A_n = \frac{8}{9} \times A_{n-1} \text{ for } n \ge 1$

Step 7 — Area behavior as n increases

The area is multiplied by $8/9$ at each step. Since $8/9$ is less than 1, the area keeps getting smaller. As $n$ increases, the area approaches 0.

Answer

(i) Stages 0 to 3 have 1, 8, 64, and 512 red squares respectively. (ii) Stage 4 has 4096 red squares. Stage 5 has 32768 red squares. (iii) Explicit formula: $t_n = 8^n$. Recursive formula: $t_0 = 1$, $t_n = 8t_{n-1}$ for $n \ge 1$. (iv) Area of red region: Stage 1 = $\frac{8}{9}$, Stage 2 = $\frac{64}{81}$, Stage 3 = $\frac{512}{729}$. Area of red region: Stage 4 = $\frac{4096}{6561}$, Stage 5 = $\frac{32768}{59049}$. Explicit formula for area: $A_n = \left(\frac{8}{9}\right)^n$. Recursive formula: $A_0 = 1$, $A_n = \frac{8}{9}A_{n-1}$ for $n \ge 1$. As $n$ increases, the area of the red region approaches 0.