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:
1. Let be the number of black triangles in Step . Define recursively.
Compare your answers:
, for .
2. Andre and Lin are asked to write an equation for that isn’t recursive. Andre writes for , while Lin writes for . 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.
Video: Learning About Sierpinski’s Triangle
Watch the following video to learn more about how to write a pattern as an nth term.
Are you ready for more?
Extending Your Thinking
Here is a geometric sequence. Find the missing terms.
3, ____, 6, ____, 12, ____, 24
Compare your answers:
, , or equivalent