Identifying Domain for a Function

4.18.2 Identifying Domain for a Function

4.18.2 • Identifying Domain for a Function

Activity

A Sierpinski triangle can be created by starting with an equilateral triangle, breaking the triangle into 4 congruent equilateral triangles, and then removing the middle triangle. Starting from a single black equilateral triangle:

$Four triangles show the stages of creating a Sierpinski triangle: the first is solid black, and each subsequent triangle has progressively smaller white triangles cut out, forming a fractal pattern.$

1. Let $S$ be the number of black triangles in Step $n$ . Define $S (n)$ recursively.

Compare your answers:

$S (1) = 3$ , $S (n) = 3 \cdot S (n - 1)$ for $n \geq 2$ .

2. Andre and Lin are asked to write an equation for $S$ that isn’t recursive. Andre writes $S (n) = 3^{n}$ for $n \geq 0$ , while Lin writes $S (n) = 3^{n} - 1$ for $n \geq 1$ . Whose equation do you think is correct? Be prepared to show your reasoning.

Compare your answers:

They are both correct. For example: Lin is correct because if the image shows Steps 1 to 4, her equation gives the correct number of triangles for each step. Andre is correct since if we start with the black triangle, which is like Step 0, then Step 1 is the one with 3 triangles and Step 2 has 9 triangles, and those numbers match his equation.

Are you ready for more?

Extending Your Thinking

Here is a geometric sequence. Find the missing terms.

3, ____, 6, ____, 12, ____, 24

Compare your answers:

$3 \sqrt{2}$ , $6 \sqrt{2}$ , $12 \sqrt{2}$ or equivalent

Video: Learning About Sierpinski’s Triangle

Watch the following video to learn more about how to write a pattern as an $n^{th}$ term.

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