A geometric sequence, starts 20, 60, . . . Define recursively and for the term.

1. An arithmetic sequence has and . Explain or show how to find the value of .
2. A geometric sequence has and . Explain or show how to find the value of .

A piece of paper has an area of 96 square inches.

Here is a growing pattern:

A growing pattern. Satge 1 has 3 branches. One is horizontal. The other two start at the end of the horizontal branch. One slants up and one slants down. Stage 2 starts with stage 1. Where the branch that slants up begins 2 more branches extend upwards and to the right. Where the branch that slants down ends, two more branches extend down and to the right. Stage 3 starts with stage 2. Where the two branches that extend upward, two additional branches extend upward from each branch. Where the two branches that extend downward, two additional branches extend downward from each branch.

1. Describe how the number of dots increases from Stage 1 to Stage 3.
2. Write a definition for sequence  so that is the number of dots in Stage .
3. Is a geometric sequence, an arithmetic sequence, or neither? Explain how you know.

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 with an area of 256 square inches, here are the first four steps:

A pattern of 4 congruent equilateral triangles. Triangle 1 is shaded. For triangle 2, connect the midpoints of each side, and remove the resulting triangle from the figure, leaving three smaller shaded congruent equilateral triangles. For triangle 3, repeat the same process and remove the 3 resulting triangles from the figure. For triangle 4, repeat the process, and remove the 9 resulting triangles from the figure.

1. Complete this table showing the number of shaded triangles in each step and the area of each triangle.

   step number	number of shaded triangles	area, in square inches, of each shaded triangle
   0	1	256
   1	3
   2
   3
   4
   5

2. Graph the number of shaded triangles as a function of the step number, then separately graph the area of each triangle as a function of the step number.
3. How are these graphs the same? How are they different?