9.1: Exploring Geometric Sequences
This activity is intended to supplement Calculus, Chapter 8, Lesson 1.
ID: 10992
Time required: 20 minutes
Activity Overview
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.
Associated Materials
- Student Worksheet: Exploring Geometric Sequences http://www.ck12.org/flexr/chapter/9733
- Transformation Graphing Application
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.
\begin{align*} \frac{2.8}{4.0} =0.7 && \frac{2.0}{2.8} \approx 0.71 && \frac{1.4}{2.0} = 0.7\end{align*}
Changing the Common Ratio
For the first part of this activity, students explore geometric sequences graphically by varying the value of \begin{align*}r\end{align*}, the common ratio. Pressing the right or left arrows will change the value of \begin{align*}r\end{align*} and updates the graph each time. Note that students will not see a graph for \begin{align*}A\end{align*} values less than zero.
They should see that \begin{align*}r-\end{align*}values between 0 and 1 could model the heights of the ball in the example.
Inquiry questions
- Why do you think the \begin{align*}r-\end{align*}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
Students will now change the equation in the Transfrm App to explore the changes in the graph of the general series, \begin{align*}a_n = a_1 \cdot r^{n - 1}\end{align*}. Students are use the up and down arrow to move from one variable to another and use the left and right arrows to change the value of the variable. They should be sure to try negative and positive values for \begin{align*}B\end{align*} for various values of \begin{align*}A\end{align*}. \begin{align*}(Y1 = B*A^{\land}(X - 1)\end{align*}
Inquiry question
- Which variable seems to have a more profound effect on the sequence? Explain.
Extension – Deriving and Applying the Partial Sum Formula
On the worksheet, students are shown the derivation of the formula for the sum of a finite geometric series. You may need to explain in detail the substitution in the third line. For example, \begin{align*}a_2 = r \cdot a_1\end{align*}, so then \begin{align*}a_3 = r \cdot a_2 = r \cdot (r \cdot a_1) = r^2 \cdot a_1\end{align*}. In the fourth line, \begin{align*}r\end{align*} is multiplied by both sides, changing \begin{align*}r^{n - 1}\end{align*} to \begin{align*}r^n\end{align*}.
Students can apply the formula to find the sum of the series given on the worksheet.
Inquiry questions:
- 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 \begin{align*}r\end{align*} is a fractional number and \begin{align*}n\end{align*} is increasing?