I. Analysis of a Spiral Representation of a Geometric Sequence

1. What geometric series is modeled by the spiral? Consider the entire area of the large square as 1 square unit. Each interior square was created by connecting consecutive midpoints by segments.

Each triangular area is a fraction of the total square area. We can express the area of the spiral as a sum of these fractions (e.g., 1/2 + 1/3 +1/4 +...). What is the partial sum of the first 6 terms? Explain how this series and its sum is represented in the geometric spiral.

2.  Model the sequence and series using your "sequence generator" Excel file.  What is the starting number? multiplier? add on?  Why do those values make sense?   If this series were to continue, to what value does this series converge? How does that value relate to the starting number, multiplier, and add on? Include a screenshot of your Excel file displaying the sequence, series, and graph.

3.  Use the Equation Editor to express the series in standard summation format (see p. 533 Theorem 9.3 from class handout).
 

Explore a similar investigation using a hexagon as the original starting figure. You will need to create your spiral in GSP.  You can start by using this 6byedge.gss script to create a hexagon.  By constructing the midpoints of each of the six sides of the hexagon and connecting them by segments, continue a similar procedure as the triangle and square to create a spiral. The shaded spiral's area is approximately what fraction of the original hexagon? Use your knowledge about infinite series to verify this sum as you did in Part I above. Use the Excel Sequence Generator to explore this series numerically and graphically.

 
III.  What if you create spiral by not using midpoints?

In each of the Baravelle spirals constructed we constructed the midpoints of each of the sides of our polygon and connected them by segments. A different type of spiral could be constructed by choosing points other than the midpoints of each of the sides of the polygon. Recursively construct a spiral that does not use the midpoints of the sides of a polygon. Write a report describing your construction and showing examples of your spirals. Do these spirals model an infinite series? If so, describe the series in formal terms.