Uploaded by MATHRoberg on 07.04.2011

Transcript:

Today we're going to talk about two really important sequences: the arithmetic and the geometric sequences.

Let's start with the arithmetic.

Ok, in the arithmetic sequence, each consecutive term differs by a constant amount.

That means if you take 2 and subtract -3, you get 5.

Let's see, 2 - (-3) is...so 2 + 3 is 5.

Let's try it with the 7 and the 2.

7 - 2 is 5.

Let's try it with the 12 and the 7.

12 - 7 is 5.

And finally, 17 - 12 is 5.

Now, notice that the difference between a term and the previous term is constant.

This is what makes an arithmetic sequence.

So this constant difference can be denoted with a d.

So the difference is d; the difference between the nth term and the term before it, which we denoted with a sub n - 1.

So what this means is if you took the third term, 7, take the third term and subtract the second term, so third term minus the second term... 0:01:32.000,0:01:34.000 That will give us d, the common difference.

Now, geometric sequences don't have a common difference.

They actually have a common ratio between a term and the previous term.

So if we take the common ratio, or if we take the ratio of the 4 to the 2 here, we have 4/2..that equals 2.

Let's try it with the 8 and the 4.

8/4...well that also equals 2.

16/8...that equals 2.

And the last one, 32/16...well, that equals 2 as well.

We call the common ratio "r."

And we could find r by taking any term, so we'll say a sub n, and dividing it by a sub n - 1.

Now, let's talk about why arithmetic sequences are so important.

Well, you could use some pretty amazing things with them.

First off, they have this general formula for finding any term in a sequence.

So, to find the nth term, we add the first term to (n-1) times our common difference.

And we also have this formula down here.

What this calculates is what is the sum of the first n terms, and you can calculate it with this formula.

Let's look at an example.

Let's start with the arithmetic.

Ok, in the arithmetic sequence, each consecutive term differs by a constant amount.

That means if you take 2 and subtract -3, you get 5.

Let's see, 2 - (-3) is...so 2 + 3 is 5.

Let's try it with the 7 and the 2.

7 - 2 is 5.

Let's try it with the 12 and the 7.

12 - 7 is 5.

And finally, 17 - 12 is 5.

Now, notice that the difference between a term and the previous term is constant.

This is what makes an arithmetic sequence.

So this constant difference can be denoted with a d.

So the difference is d; the difference between the nth term and the term before it, which we denoted with a sub n - 1.

So what this means is if you took the third term, 7, take the third term and subtract the second term, so third term minus the second term... 0:01:32.000,0:01:34.000 That will give us d, the common difference.

Now, geometric sequences don't have a common difference.

They actually have a common ratio between a term and the previous term.

So if we take the common ratio, or if we take the ratio of the 4 to the 2 here, we have 4/2..that equals 2.

Let's try it with the 8 and the 4.

8/4...well that also equals 2.

16/8...that equals 2.

And the last one, 32/16...well, that equals 2 as well.

We call the common ratio "r."

And we could find r by taking any term, so we'll say a sub n, and dividing it by a sub n - 1.

Now, let's talk about why arithmetic sequences are so important.

Well, you could use some pretty amazing things with them.

First off, they have this general formula for finding any term in a sequence.

So, to find the nth term, we add the first term to (n-1) times our common difference.

And we also have this formula down here.

What this calculates is what is the sum of the first n terms, and you can calculate it with this formula.

Let's look at an example.