# Engaging students: Geometric sequences

In my capstone class for future secondary math teachers, I ask my students to come up with ideas for engaging their students with different topics in the secondary mathematics curriculum. In other words, the point of the assignment was not to devise a full-blown lesson plan on this topic. Instead, I asked my students to think about three different ways of getting their students interested in the topic in the first place.

I plan to share some of the best of these ideas on this blog (after asking my students’ permission, of course).

This student submission comes from my former student Zachery Hasegawa. His topic, from Precalculus: geometric sequences.

How has this topic appeared in pop culture? (movies, TV, current music, video games, etc.)

Geometric sequences appear frequently in pop culture.  One example that immediately comes to mind is the movie The Happening starring Mark Wahlberg and Zoe Deschanel.  In the movie, there is a scene where a gentleman is trying to distract another woman from the chaos happening outside the jeep they’re traveling in.  He says to her “If I start out with a penny on the first day of a 31 day month and kept doubling it each day, so I’d have .01 on day 1, .02 on day 2, etc.  How much money will I have at the end of the month?” The woman franticly spouts out a wrong answer and the gentleman responds “You’d have over ten million dollars by the end of the month”.  The car goes on to crash just after that scene but as a matter of fact, you’d have exactly \$10,737,418.20 at the end of the 31-day month.  This is an example of a geometric sequence because you start out with 0.01 and to get to the next term (day), you would multiply by a common ratio of 2. What interesting things can you say about the people who contributed to the discovery and/or the development of this topic?

Geometric sequences are popularly found in Book IX of Elements by Euclid, dating back to 300 B.C.  Euclid of Alexandria, a famous Greek mathematician also considered the “Father of Geometry” was the main contributor of this theory.  Geometric sequences and series are one of the easiest examples of infinite series with finite sums.  Geometric sequences and series have played an important role in the early development of calculus, and have continued to be a main case of study in the convergence of series.  Geometric sequences and series are used a lot in mathematics, and they are very important in physics, engineering, biology, economics, computer science, queuing theory, and even finance. How can technology (specifically Khan Academy/YouTube) be used to effectively engage students with this topic:

I really like the video that Khan Academy does on YouTube about Geometric Sequences.  This particular video is a very good introduction to Geometric Sequences because he explains the difference between Geometric Sequences and Series, which I thought to be helpful because I always got the two confused with each other.  Mr. Khan starts out by explaining what exactly a Geometric Sequence is. He describes a Geometric sequence as “A progression of numbers where each successive number is a fixed multiple of the one before it.” He goes on to give numerical examples to specifically show you what he means.  He explains that a1 is typically our first term; a2 is the second term, etc.  He then explains that to get from a1 to a2, you will multiply a1 by the “common ratio” usually represented by “r. For example, “3, 12, 48, 192” is a finite geometric sequence where the common ratio, r, is 4 because to go from 3 to 12 or from 12 to 48, you multiply by 4. He goes on to explain that a Geometric Sequence is a list (sequence) of numbers (terms) that are being multiplied by a common ratio and that a Geometric Series is the sum of the terms (numbers) in the Geometric Sequence.  Using the same numbers as from the Geometric Sequence above, the geometric series is “3+12+48+192”.

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