My tutorial group has come forward with the question…
How do we tell the difference between Arithmetic and Geometric sequences?
I am going to agree, for a moment, that these two things are similar, because at first glance we will find that this comes down to rate of change, which is also known as Slope. Lets look at a typical problem in sequences.
If the 5th term in a sequence is –5 and the 15th term is –25, find the first and 50th terms.
Well, that looks just insane, but lets see what we are actually looking at. Just for explanatory purposes, lets take the term number in the sequence as X and the value it lands at as Y, which follows our understanding of x being the independent variable and y being the dependent variable.
We now have two points and from our experience, that means it is a line. Lets quickly take a moment to visualize the discussion topic though. One characteristic that sets apart the two types of sequence is the method of getting to the next term. An example of an arithmetic sequence is as follows:
1, 6, 11, 16, 21, …
The step size between each element is +5. The following is a Geometric sequence, with a step size of *5. Note the clear difference.
1, 5, 25, 125, 625, …
Something important to note is that while the steps in an arithmetic sequence simply add 5 to the previous term, a geometric sequence is multiplying the previous term by 5. From this we can deduce something that will start to clarify our situation – Arithmetic Sequences deal exclusively with Addition and Subtraction; Geometric Sequences deal with Multiplication and Division.
With each of these two example sequences we can clearly see that the step size is +5 and *5, but they stand out as green thumbs when compared to our earlier question. First offs, there are 5 elements in each sequence (Difference), secondly we have the first one to go off of (A1). A stepping stone in the pond of mathematics if you will. Lets see if we can find these two things out of the sequence by picking any two elements.
The third term in the sequence is 11, the fifth is 21. From here we can build our table of values.
The term rate of change is really a ratio of the dependent and independent variables. We talk about meters per second in physics, miles per hour in your car, chicks per dude in a party, all of them illustrate a ratio of terms. This may be reminding you of the way we found the slope of a line in earlier classes. The equations work out as follows:
Lets go ahead and work through the two points we have here.
We have done all the heavy lifting. Now we just need to figure out how to get back to the first term, and shockingly this is not much different than in previous classes when you were asked to find the Y-Intercept. Building off of what was stated earlier, we should be able to generalize this to the point that we can simply plug in a new term number and it pops out. Arithmetic sequences add to the previous term (A1).
The second term has had this difference added onto it, and the third term has had 10 added. This can be cleaned up to mean that each term beyond the first has some factor of 5 added to it and that leads us to the following formula.
With this we can solve for A1 because we have an A3.
And as you can see we have figured out the first element and can move off to the 50th term.
All of you over achievers, you should try to push forward and find the summation equations and sum the first fifty elements.
The second and fourth terms are 5 and 125, respectively. Lets follow the same pattern as before and find the difference between them.
While there are certainly a lot of means to get from 60 to five, I am certain that is not 5. This is not good folks, it looks like everything we have just tried to solve has broken down. =(
Queue the sound of the sad trombone.
But hmm, maybe the point here isn’t to stress out yet. we know that this is a different sequence, so maybe it works differently. Lets grab two other points, also two apart.
Interestingly enough, this is exactly what we needed to see to illustrate a key point. If i divide two rates here we will see something exciting happen…
OUR Rate fell out of it! Go ahead and try it with any two pairs of elements that have the same step size, including 1. No matter what you do you will find the ratio between the two to be a number that is not 1. A geometric sequence has a non linear relationship between its points, and you can depend on that.
If you take a moment to step back to the arithmetic sequence you will find that there is a similar relationship… the ratio between any two pairs is 1. This relationship is something you can depend upon and help you tell the difference between them.
You can push this further to draw the conclusion that you need to have three points in order to have the ability to dependably tell the difference between the two sequences. The third point helps you out by providing a second slope to compare and as such a second opinion about that sequence growing off of your neck.
Go ahead and see if you can derive the equation for the geometric sequence from this in a similar fashion.