 We say that a sequence is called geometric if successive terms have a common ratio. So this right here is actually a recursive definition of what we refer to as a geometric sequence. And so let's kind of unwrap what that recursive definition means. We say that a sequence is geometric. Well, you're gonna have some initial term, right? Recursive sequences start with something. So we're gonna call the first term of the sequence which of the sequences can be a sub n here. We'll just call the first term a, a is just some initial values, the seed of a recursive sequence. But if we want successive terms to have a constant ratio, what that means is this idea right here. If we take a n over a n minus one, so those would be successive terms in the sequence, right? You have the term a n minus one and it's successor a n. If we take the ratio, we should get a constant value which we're gonna call r for this, a constant ratio in this situation. So successive terms have a constant ratio if their quotient is equal to some constant number r. Now, if you take this equation and clear the denominators, well, then you'll end up with a n equals r times a n minus one. So if you know a term in the sequence, the next term in the sequence a sub n will just equal the previous term times it by this constant ratio r. And this is what we mean by geometric sequence. So some examples of that. Take the sequence two, six, 18, 54, 162. And what you're gonna see here is your first term a is two. That's just the first term in the list. That should be easy to identify here. But if you look at consecutive terms here, take six divided by two, that's equal to three. So that's gonna be our candidate for our constant ratio r. And if we look at other ratios, right? 18 divided by six, that's likewise three. If we take 54 divided by 18, that's also three. If we take 54 times 162, that likewise is gonna equal three. And so all consecutive terms in this sequence, if we divide them, will always give us three. This is evidence that our sequence is in fact geometric right here. And in fact, using this information, knowing that the common ratio is three, we can actually find the next term of the sequence. The next term of the sequence will be 162 times three, which is equal to 486. The next term of the sequence will be 486 times three, which is 1,458. The next term of the sequence will be that number times three, which is 4,374. And you can see we can keep on going, going, going. We can always find the next term of the sequence by multiple by three, times by three, times by three, times by three. Take another example. Let's consider the sequence S of N equals two to the negative, to the two to the negative N power here. So this sequence we mean would be this sequence right here. S one equals one-half, S two equals one-fourth, S three equals one-eighth, S four equals one-sixteenth, S five is one over 32, S six is equal to one over 64. So we're just taking powers of one-half. That's our sequence right here. I claim that this sequence is in fact geometric. And to show that it's geometric, we have to show that consecutive terms, their ratio is constant. So it doesn't matter what N is, you should always get a constant right here. So if you take SN over SN minus one, well SN is two to the negative N right here. SN minus one would be two to the negative N minus one power. Now because we have negative exponents, we can actually take reciprocals right here, and this will look like two to the N minus one on top, two to the N on the bottom. And then by exponent rules, because we have two N minus one on top, two N on the bottom, we can subtract the exponents like we do here. We'll get two to the N minus one minus N, in which case here the N's cancel out, we're left with two to the negative one, which is just one-half. And you'll see that this number doesn't depend on N whatsoever. It doesn't depend where we are in the sequence. This number one-half is independent of the number N. And so this is our constant ratio of one-half. This shows that this sequence is a geometric sequence whose constant ratio is one-half. Another example here, let's show that the sequence T sub N given by powers of four is likewise geometric. You're gonna see a very similar calculation here. If we look at the ratio of consecutive terms, you're gonna take TN on top, TN minus one on the bottom. So for TN, you're just gonna record the formula just straight from above four to the N. For TN minus one, you're gonna replace in the formula the N with an N minus one, like we see right here. And that's your ratio. Simplifying that, you can subtract the powers. You get four to the N minus N minus one. And then the N's will cancel because you get N minus N. Then you're gonna get a negative negative one. So you end up with four to the first, that is four. And that's again our constant ratio. This sequence would look like four, 16, two, sorry, 256. We're just looking at powers of four right here. And so these last two examples you'll notice, right? We had powers of one half, powers of four. And then on the first example, right? We were increasing by powers of three. It turns out that geometric sequences, in some ways, behave like exponentials. And we actually see this in general. So for a general geometric sequence right here, let's say the initial term is A and its constant ratio is R. So therefore, the first term will just be A. It's the initial term. To find the second term, A2, we're gonna multiply the first term A1 by R. But as A1 is just an A, A2 will just look like R times A. For A3, we have to times R, or A2 by R to get A3. But since A2 was just equal to R times A, we could insert that here. We get R times RA, which is R squared A. So we do this for A4. A4 will be R times A3. A3 is R squared times A. Therefore, A to the fourth, sorry, A4 will be R cubed A. If we do this for the fifth one, right? A5 will be R times A4. A4 we saw was R cubed A. And so you get another R. You get R squared to the A. And so by mathematical induction, we can follow this pattern and see the following. A to the N will just equal R times A to the N minus one. It's predecessor. But by induction, A to the N minus one will be R in minus two power times A. And so you increase the power by one, you're going to get R to the N minus one times A. In which case, we then see in general that a geometric sequence is very much like an exponential expression. You're going to have some initial value, A. And then that's going to be multiplied by some power of R. But our power of R is going to be one less than our current position. So the Nth spot will have the N minus one power of R. And that defines a general geometric sequence.