Hi, and welcome. My name is Anthony Varela. And today I'm going to introduce geometric sequences. We'll talk about how a geometric sequence has a common ratio between its terms. We will write a formula for a given sequence, and then use the formula to find the value of the nth term.
So a geometric sequence is a set of numbers in numerical order with a non-zero common ratio between each term. So here's an example of a geometric sequence. We have the terms 2, 6, 18, 54, so on and so forth.
And there's a common ratio between each term. So what's the common ratio? Well, it's the ratio between any two consecutive terms in a geometric sequence. It's a constant value.
So for example, to get from 2 to 6, we multiply by 3. And we multiply by 3 again to get the value of the next term. And 18 times 3 is 54.
So the common ratio then is 3. It's the value that we have to multiply to get from one term to the next. So a geometric sequence is a set of numbers with a constant value multiplied from one term to the next.
So now that we know that the common ratio then to this geometric sequence is 3, what are the values of the next two terms? Well, I can take the last term that we know, 54, and multiply it by 3. So the value after 54 is 162. And the term that comes after 162 would be 3 times 162-- 486.
So before we get to a formula for geometric sequences, I'd like to talk about divergent and convergent sequences. So the example that we just worked with is an example of a divergent sequence. It's a sequence whose terms do not have a finite limit. They tend toward positive or negative infinity.
So because our common ratio was 3, we're constantly multiplying a term by 3 to get the value of the next term. So that's going to approach positive infinity.
Now even if our common ratio was a negative number, that absolute value of our term would still be getting larger and larger. It would just be alternating between positive and negative.
So with a divergent sequence, the terms have no limits. And this is characteristic when the common ratio r, the absolute value of r, is greater than 1. So common ratios that are greater than 1 or less than negative 1 would be divergent sequences.
Let's take a look at this sequence. Here we have 32, 16, 8, 4, 2, so on and so forth. So we're cutting our terms in half to get the value of the next term. And this is a convergent sequence.
So it's a sequence whose terms have a finite limit. They tend toward a specific value. So if we're always cutting a value in half, we're getting closer and closer and closer to 0. So that's the value that our terms converge to.
And this is characteristic then when the absolute value of the common ratio is less than 1. So if your common ratio is in between negative 1 and positive 1, you have a convergent sequence.
Now when finding the value of a term in a geometric sequence, we saw earlier that it was easy enough to just multiply a 54 by 3 to get 162, and multiply it by 3 again to get 486.
But what if I wanted to find the value of the 100th term? I don't want to start at 2 and multiply it by 3 a whole bunch of times. So we're going to use a formula.
And our formula to find the value of the nth term, so I'll write this as a sub n, equals the value of your first term, a sub 1, multiplied by your common ratio r raised to the power of n minus 1, where n is the number of the term.
And the reason why you raise this to the power of n minus 1 is because, for example, to find the value of 6, we started with two and multiplied it by 3 one time. To find the value of the third term, we started with the first term, multiplied it by our common ratio two times. So it's n minus 1 is the exponent there.
So let's use this formula for the value of a term in a geometric sequence. So to write the formula for this specific geometric sequence, we say that the value of the nth term equals the first term, so that would be 2, multiplied by the common ratio.
So what's the common ratio? How I like to find the common ratio is take the second term and divide it by the first term, so 3 divided by 2. So that would be 1 and 1/2.
And we can check to make sure that that is indeed the common ratio. So I'm going to interpret 1 and 1/2 as being 3/2. So I can multiply then a 3 by 3/2 to get 9/2. And we see that our numerator is being multiplied by 3 and our denominator is being multiplied by 2. So our common ratio is indeed 3/2, or 1 and 1/2.
And then we need to raise this then to the power of n minus 1. So now that we have a formula, let's find the value of the eighth term. So the value of the eighth term then equals 2 times 1.5 to the 7th power.
So we need to evaluate this exponent first. So 1.5 to the 7th power is approximately 17.086. Multiply that then by 2, so the value of the eighth term is approximately 34.17.
Now what if we wanted to find the term that has a value of 22.78. So here we know what a sub n is. We just need to figure out what n is.
So the first thing that I'm going to do here is divide the equation by 2. That's because this exponent applies to 1.5, not to 2. So I can get rid of that through division. So I have that 11.39 equals 1.5, raised to the power of n minus 1.
To take care of this exponent of n minus 1, I'm going to take the log of both sides. So I have enclosed this side of the equation, inside of a log, and this entire expression here inside of a log.
Well, say that the log of 11.39 is about 1.057. And we have an exponent inside of a log operation. So we can bring that out. So I have the quantity n minus 1 being multiplied by the log of 1.5, which I'll say is 0.176.
Well, now I can isolate my quantity n minus through division. And I'll round to the nearest integer because I know that n has to be an integer. So I have that 6 equals n minus 1, which means that the seventh term has a value of 22.78.
So let's review our lesson on this introduction to geometric sequences. A geometric sequence is a set of numbers with a constant value multiplied from one term to the next. We talked about divergent sequences-- they have no limit to their terms-- and convergent sequences that do have a limit to the term.
And we used this formula to find the value of the nth term. And it involves the value of the nth the term, the value of the first term, our common ratio raised to an exponent power, n minus.
Thanks for watching this introduction to geometric sequences. Hope to see you next time.