The document describes several examples of iterative sequences: 1. Connecting the midpoints of triangles to generate smaller nested triangles with each iteration. 2. Alternating between connecting midpoints and removing coloring in triangles to produce a pattern of blank and shaded triangles. 3. Continuing a sequence of nested figures by applying the same transformation at each step. 4. Extending a numeric sequence following a given rule to generate additional terms. 5. Building up a sequence of letters by appending the next letter and repeating the prior term.

1. Drawing Iterations
1. In the figure below, we get the first iteration by connecting the midpoints of the sides of the triangle. Note this
creates a smaller, rotated triangle in iteration #1. Now repeat the process, using the smaller rotated triangle as
your starting point to create iteration #2. Continue to create #3 and #4.
2. Draw iteration #3 in the sequence below.
3. Below you are given the first two iterations. Note there are two steps to the iteration this time – first connect
the midpoints of the three sides to form a new, smaller triangle. Then remove the coloring from this new
triangle so that there are three small black trangles. These three small blank trangles are your new starting
point. You should connect midpoints in each of them and “erase” the middle triangle. Continue this, creating
iterations #2, 3, and 4.
4. Below you are given iterations #0, 1, and 2 of a sequence. See if you can figure out what you do to each figure to
get the next, and then try to continue to find iterations #3 and 4.
MAD Spring 2010 AVC

2. 5. Below is a “Fibonacci-type” sequence that starts with 1 and 3. Write down 6 more iterations of this sequence.
1, 3, 4, …
6. Below is a sequence of letters that is iterated.
Start with an “a.”
Iteration #0: a
To grow the pattern, add the next letter in the alphabet and then repeat the previous step before and after (which is
just the letter “a” in this case.) The next step, then, is “aba.” The new letter is in bold below.
Iteration #1: aba
Continue by adding the next letter, “c,” and repeating the previous step before and after. The new letter is in bold
below.
Iteration #2: abacaba
The fourth iteration adds the letter “d” and repeats the pattern. Write down iterations #4, 5, 6.
MAD Spring 2010 AVC