Video transcript
What I want to do in thisvideo is familiarize ourselves with the notion of a sequence. And all a sequence is isan ordered list of numbers. So for example, I couldhave a finite sequence-- that means I don't have aninfinite number of numbers in it-- where, let's say, Istart at 1 and I keep adding 3. So 1 plus 3 is 4. 4 plus 3 is 7. 7 plus 3 is 10. And let's say I only have thesefour terms right over here. So this one we wouldcall a finite sequence. I could also have aninfinite sequence. So an example of aninfinite sequence-- let's say we start at3, and we keep adding 4. So we go to 3, to 7, to 11, 15. And you don't always haveto add the same thing. We'll explore fancier sequences. The sequences where youkeep adding the same amount, we call thesearithmetic sequences, which we will alsoexplore in more detail. But to show thatthis is infinite, to show that we keep thispattern going on and on and on, I'll put three dots. This just means we're going tokeep going on and on and on. So we could call thisan infinite sequence. Now, there's a bunchof different notations that seem fancy fordenoting sequences. But this is all they refer to. But I want to make uscomfortable with how we can denote sequences andalso how we can define them. We could say thatthis right over here is the sequence a sub kfor k is going from 1 to 4, is equal to thisright over here. So when we lookat it this way, we can look at each of these asthe terms in the sequence. And this right over herewould be the first term. We would call that a sub 1. This right over herewould be the second term. We'd call it a sub 2. I think you get thepicture-- a sub 3. This right over here is a sub 4. So this just says, all of thea sub k's from k equals 1, from our first term, allthe way to the fourth term. Now, I could also define itby not explicitly writing the sequence like this. I could essentially doit defining our sequence as explicitly using kind of afunction notation or something close to function notation. So the same exactsequence, I could define it as a sub k from k equals 1 to4, with-- instead of explicitly writing the numbershere, I could say a sub k is equal to some function of k. So let's see what happens. When k is 1, we get 1. When k is 2, we get 4. When k is 3, we get 7. So let's see. When k is 3, we added 3 twice. Let me make it clear. So this was a plus 3. This right overhere was a plus 3. This right overhere is a plus 3. So whatever k is,we started at 1. And we added 3 one lessthan the k term times. So we could say that thisis going to be equal to 1 plus k minus 1times 3, or maybe I should write 3 times kminus 1-- same thing. And you can verifythat this works. If k is equal to 1, you'regoing to get 1 minus 1 is 0. And so a sub 1 is going to be 1. If k is equal to 2, you're goingto have 1 plus 3, which is 4. If k is equal to 3, youget 3 times 2 plus 1 is 7. So it works out. So this is one way to explicitlydefine our sequence with kind of this function notation. I want to make it clear--I have essentially defined a function here. If I wanted a moretraditional function notation, I could have writtena of k, where k is the term thatI care about. a of k is equal to 1plus 3 times k minus 1. This is essentiallya function, where an allowable input,the domain, is restricted to positive integers. Now, how would I denote thisbusiness right over here? Well, I could say thatthis is equal to-- and people tend to use a. But I could use the notationb sub k or anything else. But I'll do a again-- a sub k. And here, we're goingfrom our first term-- so this is a sub 1,this is a sub 2-- all the way to infinity. Or we could define it-- if wewanted to define it explicitly as a function-- we could writethis sequence as a sub k, where k starts at the firstterm and goes to infinity, with a sub k is equaling--so we're starting at 3. And we are adding4 one less time. For the second term,we added 4 once. For the third term,we add 4 twice. For the fourth term,we add 4 three times. So we're adding 4 one lessthan the term that we're at. So it's going to beplus 4 times k minus 1. So this is anotherway of defining this infinite sequence. Now, in both of thesecases, I defined it as an explicit function. So this right overhere is explicit. That's not an attractive color. Let me write this in. This is an explicit function. And so you mightsay, well, what's another way of definingthese functions? Well, we can also defineit, especially something like an arithmetic sequence, wecan also define it recursively. And I want to be clear-- notevery sequence can be defined as either an explicitfunction like this, or as a recursive function. But many can,including this, which is an arithmeticsequence, where we keep adding the samequantity over and over again. So how would we do that? Well, we could also--another way of defining this first sequence,we could say a sub k, starting at k equals1 and going to 4 with. And when you define asequence recursively, you want to define what yourfirst term is, with a sub 1 equaling 1. You can define every other termin terms of the term before it. And so then wecould write a sub k is equal to the previous term. So this is a sub k minus 1. So a given term is equalto the previous term. Let me make it clear-- this isthe previous term, plus-- in this case, we'readding 3 every time. Now, how does this make sense? Well, we're definingwhat a sub 1 is. And if someone says, well,what happens when k equals 2? Well, they're saying, well, it'sgoing to be a sub 2 minus 1. So it's going tobe a sub 1 plus 3. Well, we know a sub 1 is 1. So it's going to be1 plus 3, which is 4. Well, what about a sub 3? Well, it's going to bea sub 2 plus 3. a sub 2, we just calculated as 4. You add 3. It's going to be 7. This is essentiallywhat we mentally did when I first wrote out thesequence, when I said, hey, I'm just going to start with 1. And I'm just going to add 3for every successive term. So how would we do this one? Well, once again, we couldwrite this as a sub k. Starting at k, thefirst term, going to infinity with-- ourfirst term, a sub 1, is going to be 3, now. And every successiveterm, a sub k, is going to be the previousterm, a sub k minus 1, plus 4. And once again, you start at 3. And then if you wantthe second term, it's going to be thefirst term plus 4. It's going to be 3 plus 4. You get to 7. And you keep adding 4. So both of these,this right over here is a recursive definition. We started withkind of a base case. And then every term isdefined in terms of the term before it or in termsof the function itself, but the function fora different term.
