This activity is intended to supplement Calculus, Chapter 8, Lesson 1.
Time required: 20 minutes
In this activity, students will explore geometric series. They will consider the effect of the value for the common ratio and first term using the TRANSFRM APP for the TI-84. Students will graphically analyze geometric series using graphs.
Topic: Sequences and Series
Explore geometric sequences
Teacher Preparation and Notes
This activity serves as a nice introduction to geometric series. Students will need the TRANSFRM APP on their TI-84. The accompanying worksheet can add depth to the questions and helpful key press explanation.
An extension would be to solve infinite geometric series that converge using sigma notation and the limit of the partial sum formula.
Students should make sure to turn off all Stat Plots before beginning the activity.
To exit the Transformation App, go to the App menu, press enter on Transfrm and then select Uninstall.
To download the Transformation Graphing Application, go to http://www.education.ti.com/calculators/downloads/US/Activities/Detail?id=10992 and select Transformation Graphing from the Applications header.
Example of Geometric Sequence
Students are shown the path of a ball that is bouncing. Show that the common ratio of the heights is approximately the same.
Changing the Common Ratio
- Why do you think the r−value is called the common ratio? [Answer: The ratios of consecutive terms are the same; They have that in common.]
- What would happen if you added all the terms of this sequence? For what common ratio conditions do you think the sum will diverge (get larger, and not converge to some number)?
Changing the Initial Value and the Common Ratio
- Which variable seems to have a more profound effect on the sequence? Explain.
Extension – Deriving and Applying the Partial Sum Formula
Students can apply the formula to find the sum of the series given on the worksheet.
- If you cut a piece of paper in half, then cut one of the halves in half and repeated the process, what is the sum of all these fractional pieces? [Answer: one whole]
- However, if you gave your teacher one penny on the first day of class, 2 on the next, then 4, 8, 16, 32, ... would this number converge?
- How would the formula for a finite geometric sequence look if r is a fractional number and n is increasing?