About this video
Using 1, 1/2, 1/4, 1/8, ... as an example, the video defines a geometric sequence and its common ratio, and writes the nth term as a_n = (1/2)^(n-1). It then explains that a geometric series is the sum of a geometric sequence and shows how to write it in sigma notation.
You'll be able to
- identify the common ratio of a geometric sequence.
- write the nth term of a geometric sequence as a_n = a x r^(n-1).
- explain the difference between a sequence and a series.
Key words
- Geometric sequence
- A list of numbers where each term is the previous term multiplied by the same number.
- Common ratio
- The fixed number you keep multiplying by to get from one term to the next.
- Geometric series
- The sum of the terms of a geometric sequence.
- Infinite
- Going on forever, with no last term.
Transcript
Let's say that I have a geometric series. A geometric sequence, I should say. We'll talk about series in a second. So a geometric series, let's say it starts at 1, and then our common ratio is 1/2. So the common ratio is the number that we keep multiplying by.
So 1 times 1/2 is 1/2, 1/2 times 1/2 is 1/4, 1/4 times 1/2 is 1/8, and we can keep going on and on and on forever. This is an infinite geometric sequence. And we can denote this. We can say that this is equal to the sequence of a sub n from n equals 1 to infinity, with a sub n equaling 1 times our common ratio to the n minus 1.
So it's going to be our first term, which is just 1, times our common ratio, which is 1/2. 1/2 to the n minus 1. And you can verify it. This right over here you can view as 1/2 to the 0 power. This is 1/2 half to the first power, this is 1/2 squared. 1/2 to the first, this is 1/2 squared.
So the first term is 1/2 to the 0. The second term is 1/2 to the 1. The third term is 1/2 squared. So the nth term is going to be 1/2 to the n minus 1. So this is just really 1/2 to the n minus 1 power. Fair enough. Now, let's say we don't just care about looking at the sequence.
We actually care about the sum of the sequence. So we actually care about not just looking at each of these terms, see what happens as I keep multiplying by 1/2, but I actually care about summing 1 plus 1/2 plus 1/4 plus 1/8, and keep going on and on and on forever.
So this we would now call a geometric series. And because I keep adding an infinite number of terms, this is an infinite geometric series. So this right over here would be the infinite geometric series. A series you can just view as the sum of a sequence. Now, how would we denote this?
Well, we can use summing notation. We could say that this is equal to the sum. We could say that this is equal to the sum. Let me make sure I'm not falling off the page. Let me just scroll over to the left a bit. The sum from n equals 1 to infinity of a sub n.
And a sub n is just 1/2 to the n minus 1. 1/2 to the n minus 1 power. So you just say OK, when n equals 1, it's 1/2 to the 0, which is 1. Then I'm going to sum that to when n equals 2, which is 1/2, when n equals 3, it's 1/4. On and on, and on, and on. So all I want to do in this video is to really clarify differences between sequences and series, and make you a little bit comfortable with the notation.
In the next few videos, we'll actually try to take sums of geometric series and see if we actually get a finite value.
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