June 2010 - Page 4 of 6 - Math and Multimedia

GeoGebra Tutorial 19 – Basic Spreadsheet Recording

June 11, 2010 GB GeoGebra, Software Tutorials

This is the 19th tutorial of the GeoGebra Intermediate Tutorial Series. If this is your first time to use GeoGebra, you might want to read first the GeoGebra Essentials Series.

A spreadsheet is a program that can be used to organize data in tables and perform mathematical computations. Recently, GeoGebra integrated spreadsheet in its graphical user interface. In this tutorial, we learn how to use the GeoGebra spreadsheet.

Figure 1

The figure above shows the different parts of a spreadsheet. The following are the descriptions. You should familiarize yourself with these terms because we are going to use them in this tutorial and the two more tutorials to come. » Read more

6 comments formula in GeoGebra spreadsheet, GeoGebra Record to Spreadsheet tool, geogebra slider, GeoGebra spreadsheet, GeoGebra Tutorial Series, parts of a spreadsheet

GeoGebra Short-Term Course Concluded

June 9, 2010 GB Announcements, Miscellaneous

Last Saturday, June 5, 2010, we have concluded our 3-Saturday seminar on Using GeoGebra in Teaching Mathematics. Below are some of the pictures taken from the said short-term course.

That’s me on the first pic explaining the mathematics behind the “how-to” construction. If you’re wondering why our whiteboard is high, well, I am just short. I am 5’3” feet, or maybe shorter (chuckles).

As you can see, the participants are really serious. 🙂

One of our participants constructing her GeoGebra applet.

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So if you want to invite me* to conduct a lecture on GeoGebra in your school, I am just an email away.

*Philippines only (unless, of course, your school is willing to shoulder my expenses. Just kidding 🙂 ).

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Geometric Sequences and Series

June 9, 2010 GB Calculus and Analysis, College Mathematics, High School Algebra, High School Mathematics

Introduction

We have discussed about arithmetic sequences, its characteristics and its connection to linear functions. In this post, we will discuss another type of sequence.

The sequence of numbers 2, 6, 18, 54, 162, … is an example of an geometric sequence. The first term 2 is multiplied by 3 to get the second term, the second term is multiplied by 3 to get the third term, the third term is multiplied by 3 to get the fourth term, and so on. The same number that we multiplied to each term is called the common ratio. Expressing the sequence above in terms of the first term and the common ratio, we have 2, 2(3), 2(3²), 2(3³), …. Hence, a geometric sequence, also known as a geometric progression, is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed non-zero number called the common ratio.

The Sierpinski triangle below is an example of a geometric representation of a geometric sequence. The number of blue triangles, the number of white triangles, their areas, and their side lengths form different geometric sequences. It is left to the reader, as an exercise, to find the rules of these geometric sequences.

Figure 1 - The Seriepinski Triangles.

To generalize, if a₁ is its first term and the common ratio is r, then the general form of a geometric sequence is a₁, a₁r, a₁r², a₁r³,…, and the n^th term of the sequence is a₁r^n-1.

A geometric series, on the other hand, is the sum of the terms of a geometric sequence. Given a geometric sequence with terms a₁r, a₁r², a₁r³,…, the sum S_nof the geometric sequence with n terms is the geometric series a₁ + a₁r + a₁r², a₁r³ + … + ar^n-1. Multiplying S_n by -r and adding it to S_n, we have

Hence, the sum of a geometric series with n terms, and $r \neq 1 = \displaystyle\frac{a_1(1-r^n)}{1-r}$ .

Sum of Infinite Geometric Series and a Little Bit of Calculus

Note: This portion is for those who have already taken elementary calculus.

The infinite geometric series $\{a_n\}$ is the the symbol $\sum_{n=1}^\infty a_n = a_1 + a_2 + a_3 + \cdots$ . From above, the sum of a finite geometric series with $n$ terms is $\displaystyle \sum_{k=1}^n \frac{a_1(1-r^n)}{1-r}$ . Hence, to get the sum of the infinite geometric series, we need to get the sum of $\displaystyle \sum_{n=1}^\infty \frac{a_1(1-r^n)}{1-r}$ . However, $\displaystyle \sum_{k=1}^\infty \frac{a_1(1-r^n)}{1-r} = \lim_{n\to \infty} \frac{a_1(1-r^n)}{1-r}$ .

Also, that if $|r| < 1$ , $r^n$ approaches $0$ (try $(\frac{2}{3})^n$ or any other proper fraction and increase the value of $n$ ), thus, $\displaystyle \sum_{n=1}^\infty \frac{a_1(1-r^n)}{1-r} = \lim_{n \to \infty} \frac{a_1(1-r^n)}{1-r} = \frac{a_1}{1-r}$ . Therefore, sum of the infinite series $\displaystyle a_1 + a_2r + a_2r^2 + \cdots = \frac{a_1}{1-r}$ .

One very common infinite series is $\displaystyle \sum_{n=1}^{\infty} \frac{1}{2n} = \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \frac{1}{16} + \cdots$ , or the sum of the areas of the partitions of the square with side length 1 unit shown below. Using the formula above,

$\displaystyle \sum_{n=1}^{\infty} \frac{1}{2n} = \frac{a_1}{1-r} = \frac{\frac{1}{2}}{1-\frac{1}{2}} = 1$ .

Figure 2 - A representation of an infinite geometric series.

This is evident in the diagram because the sum of all the partitions is equal to the area of a square. We say that the series $\displaystyle \sum_{n=1}^{\infty} \frac{1}{2n}$ converges to 1.

8 comments common ratio, geometric progressions, geometric sequence, infinite geometric sequence, sierpinski triangle, sum of infinite geometric series