Note: There will be vitalized examples throughout these articles, so make sure you have the plug-in plugged... in! If you're using IE, a link should pop up when you try to run the example, but for all you Firefox or other browser users, here's the link: http://www.clickteam.com/vitalize4/plugin.html

Intro

Ok, welcome to this series of articles about Arithmetic Progressions (APs) and Geometric Progressions (GPs). Most of you are probably still wondering what on Earth these things are, but trust me, you'll be an expert on them after this! Basically they are different kinds of number patterns or sequences that can be useful in many different ways, if you know how to use them of course!

I'll start by explaining the basic properties of a Geometric Progression, as they are the easier of the two to understand. Please read on...

GEOMETRIC PROGRESSIONS

First of all, let me show you an example of a Geometric Progression:

2, 6, 18, 54, 162, ...

Now it may just look like a series of random numbers, but there are also a few important things to note. First thing, note how the GP starts at 2. GPs can start at any number except 0, positive or negative, integer or fraction, etc. Second thing, and this is probably the most important one, as it defines the sequence as a GP; the numbers are being multiplied by 3 each time, eg:
2*3=6
6*3=18
18*3=54
You get the picture. A GP must have a Common Ratio by which you can determine the next number in the sequence; so in this case the Common Ratio is 3. The common ratio can be negative or positive, integer or fraction, but not 0. It is often referred to as a lower-case letter r (for ratio).
The third thing to notice is that the numbers in the progression will keep getting larger and larger, towards positive infinity. The progression never ends.

The fourth and last thing you should know is that each number in a GP is called a term. For example the first term is the first number, the seventh term is the seventh number, etc.. Instead of writing term, we can shorten it to just a lower-case letter t with a number after it, eg: t1, or t7.

Ok, now that we've examined a GP, we can generate some rules for all GPs.

RULES FOR A GEOMETRIC PROGRESSION

Rule 1: A GP must start on any number other than 0.
Rule 2: A GP must have a Common Ratio (r) that can be any number other than 0.
Rule 3: A GP can (but not always) progress towards infinity.
Rule 4: Each number in a GP is called a term.

Easy as 1,2,3! ...4

Now we've got the rules that define a Geometric Progression. However, there are still a few interesting things about the Common Ratio that come into play.

Note: To find the Common Ratio of a GP, simply divide a term by the term immediately before it.

THE SEVEN PROPERTIES OF THE COMMON RATIO (r)

1. If r is positive, every term in the progression will be the same sign as the first term:
3, 6, 12, 24, 48, ... r = 2
-4, -12, -36, -108, -324, ... r = 3


2. If r is negative, then the terms will alternate between positive and negative:
6, -12, 24, -48, 96, ... r = -2

3. If r is greater than 1, there will be exponential growth towards positive infinity.

4. If r is 1, the term will be constant:
7, 7, 7, 7, 7, ... r = 1

5. If r is between -1 and 1, but not 0, there will be exponential decay towards 0:
3, 1.5, 0.75, 0.375, 0.1875, ... r = 0.5

6. If r is -1, the term will be constant, but alternating between positive and negative:
-5, 5, -5, 5, -5, ... r = -1

7. If r is lower than -1, there will be exponential growth towards infinity.

And there you have it. The Common Ratio will effect the GP in different ways.

At this point, you know all the rules that define a GP, you know all the different effects that the Common Ratio can have, so it is time to move on to the basic formula of a GP.

FINDING THE NTH TERM

Without knowing any values of a GP, it can be represented in it's simplest form as:

a, a*(r), a*(r*r), a*(r*r*r), a*(r*r*r*r), ...

Where a is the first term of the sequence, and r is of course, the common ratio.
If you're good with maths, you may have already noticed that this can be simplified to:

a, a*r, a*r^2, a*r^3, a*r^4, ...
Much better!

Now imagine you are trying to find the Seventh Term (t7) of this GP:

2, 6, 18, 54, 162, ...

You could just keep multiplying by 3 until you get to the seventh number, but this is an impractical method to use, especially if you were trying to find something like t43!

So instead, let's look at our GP formula so far:

a, a*r, a*r^2, a*r^3, a*r^4, a*r^5, a*r^6, ...

Now if you count the terms along, you will find that the seventh term is a*(r^6). Now if we substitute in the values from our GP, we get:
2*3^6

= 2*729
= 1458

And behold, 1458 is the seventh term of the GP!

The final thing we need to do to make the formula complete is to make it universal, which means we can use one small equation to find any term of the GP. Notice how when finding the seventh term, r had a power of 6, which is 7-1. Infact, to find any term of any GP, the almighty formula is:

tn = a * r^(n-1)

Where n is the term number that you wish to find. For example, to find t7 we used:

a*r^6

Which can also be written as:

a*r^(7-1)

Which proves the formula to be true.

Similarly, to find t43 of the GP 2, 6, 18, 54, 162, ... we simply go:

2 * 3^(43-1)

Which, if you punch into a calculator, just so happens to be 218837978263024718418.

So this formula is handy when you want to know a specific term of a GP.

Now I'd like to show you a quick vitalized example of the GP formula in action. First, define the GP by entering the first term and the Common Ratio, then you can find any term in the sequence; it will be calculated using the tn = a * r^(n-1) formula. Remember, the first term and Common Ratio don't have to be whole numbers, try experimenting with the Seven Properties of the Common Ratio as described above.



And there you have it.

This is basically all you need to know about GPs at the moment, so it seems like a good place to end this article. In future articles I will cover Arithmetic Progressions, and a new formula that enables you find the sum of the first n terms of a GP or AP.

So that's all for now! Please, if you don't understand something or find that this whole article might as well have been written in Alien Hieroglyphs for all the sense it made, don't hesitate to ask a question or leave a comment!