Question 3

We will find the remaining area at each step by observing how much of the shape is kept.

Step 1 — Sierpinski's Carpet: Understanding the pattern

Let us imagine a square. We call this Step 0. Its area is 1 square unit. To get to Step 1, we divide this square into 9 smaller, equal squares. Then, we remove the central square. This means 1 out of 9 squares is removed. So, 8 out of 9 squares remain. The remaining area is $\frac{8}{9}$ of the original area.

$\text{Area at Step 0} = 1 \text{ sq. unit}$

$\text{Area at Step 1} = \frac{8}{9} \times \text{Area at Step 0}$

$= \frac{8}{9} \times 1$

$\boxed{\frac{8}{9} \text{ sq. units}}$

Step 2 — Sierpinski's Carpet: Finding the area at Step 2

Now, let us look at Step 2. We take each of the 8 remaining squares from Step 1. For each of these 8 squares, we repeat the process. We divide each into 9 smaller squares and remove the central one. So, each of these 8 squares now has $\frac{8}{9}$ of its own area remaining. The total remaining area will be $\frac{8}{9}$ of the area from Step 1.

$\text{Area at Step 2} = \frac{8}{9} \times \text{Area at Step 1}$

$= \frac{8}{9} \times \frac{8}{9}$

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

$\boxed{\frac{64}{81} \text{ sq. units}}$

Step 3 — Sierpinski's Carpet: Generalizing to the $n$th step

We can see a pattern forming here. At each step, the remaining area is multiplied by $\frac{8}{9}$. So, for the $n$th step, we multiply $\frac{8}{9}$ by itself $n$ times. This is written as $\left(\frac{8}{9}\right)^n$.

$\text{Area at Step n} = \left(\frac{8}{9}\right) \times \left(\frac{8}{9}\right) \times \dots \times \left(\frac{8}{9}\right) \quad (n \text{ times})$

$\boxed{\left(\frac{8}{9}\right)^n \text{ sq. units}}$

Step 4 — Sierpinski's Gasket: Understanding the pattern

Let us look at the triangle shown in the image. We call this Step 0. Its area is 1 square unit. To get to Step 1, we find the midpoints of each side of the triangle. We connect these midpoints to form a new, inverted triangle in the center. This divides the original triangle into 4 smaller, equal triangles. We remove the central inverted triangle. This means 1 out of 4 triangles is removed. So, 3 out of 4 triangles remain. The remaining area is $\frac{3}{4}$ of the original area.

$\text{Area at Step 0} = 1 \text{ sq. unit}$

$\text{Area at Step 1} = \frac{3}{4} \times \text{Area at Step 0}$

$= \frac{3}{4} \times 1$

$\boxed{\frac{3}{4} \text{ sq. units}}$

Step 5 — Sierpinski's Gasket: Finding the area at Step 2

Now, let us look at Step 2. We take each of the 3 remaining triangles from Step 1. For each of these 3 triangles, we repeat the process. We divide each into 4 smaller triangles and remove the central one. So, each of these 3 triangles now has $\frac{3}{4}$ of its own area remaining. The total remaining area will be $\frac{3}{4}$ of the area from Step 1.

$\text{Area at Step 2} = \frac{3}{4} \times \text{Area at Step 1}$

$= \frac{3}{4} \times \frac{3}{4}$

$= \left(\frac{3}{4}\right)^2$

$\boxed{\frac{9}{16} \text{ sq. units}}$

Step 6 — Sierpinski's Gasket: Generalizing to the $n$th step

We can see a pattern forming here too. At each step, the remaining area is multiplied by $\frac{3}{4}$. So, for the $n$th step, we multiply $\frac{3}{4}$ by itself $n$ times. This is written as $\left(\frac{3}{4}\right)^n$.

$\text{Area at Step n} = \left(\frac{3}{4}\right) \times \left(\frac{3}{4}\right) \times \dots \times \left(\frac{3}{4}\right) \quad (n \text{ times})$

$\boxed{\left(\frac{3}{4}\right)^n \text{ sq. units}}$

Answer

(a) For Sierpinski's Carpet, the area of the remaining region after the $n$th step is $\left(\frac{8}{9}\right)^n$ sq. units. (b) For Sierpinski's Gasket, the area of the remaining region after the $n$th step is $\left(\frac{3}{4}\right)^n$ sq. units.