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  • What I want to do in this video is familiarize ourselves

  • with the notion of a sequence.

  • And all a sequence is is an ordered list of numbers.

  • So for example, I could have a finite sequence--

  • that means I don't have an infinite number of numbers

  • in it-- where, let's say, I start 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 these four terms right over here.

  • So this one we would call a finite sequence.

  • I could also have an infinite sequence.

  • So an example of an infinite sequence--

  • let's say we start at 3, and we keep adding 4.

  • So we go to 3, to 7, to 11, 15.

  • And you don't always have to add the same thing.

  • We'll explore fancier sequences.

  • The sequences where you keep adding the same amount,

  • we call these arithmetic sequences,

  • which we will also explore in more detail.

  • But to show that this is infinite,

  • to show that we keep this pattern going on and on and on,

  • I'll put three dots.

  • This just means we're going to keep going on and on and on.

  • So we could call this an infinite sequence.

  • Now, there's a bunch of different notations

  • that seem fancy for denoting sequences.

  • But this is all they refer to.

  • But I want to make us comfortable with how

  • we can denote sequences and also how we can define them.

  • We could say that this right over here

  • is the sequence a sub k for k is going from 1 to 4,

  • is equal to this right over here.

  • So when we look at it this way, we

  • can look at each of these as the terms in the sequence.

  • And this right over here would be the first term.

  • We would call that a sub 1.

  • This right over here would be the second term.

  • We'd call it a sub 2.

  • I think you get the picture-- a sub 3.

  • This right over here is a sub 4.

  • So this just says, all of the a sub k's from k equals 1,

  • from our first term, all the way to the fourth term.

  • Now, I could also define it by not explicitly writing

  • the sequence like this.

  • I could essentially do it defining our sequence

  • as explicitly using kind of a function notation or something

  • close to function notation.

  • So the same exact sequence, I could define it

  • as a sub k from k equals 1 to 4, with-- instead of explicitly

  • writing the numbers here, 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 over here was a plus 3.

  • This right over here is a plus 3.

  • So whatever k is, we started at 1.

  • And we added 3 one less than the k term times.

  • So we could say that this is going to be equal to 1

  • plus k minus 1 times 3, or maybe I

  • should write 3 times k minus 1-- same thing.

  • And you can verify that this works.

  • If k is equal to 1, you're going 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 going to have 1 plus 3, which is 4.

  • If k is equal to 3, you get 3 times 2 plus 1 is 7.

  • So it works out.

  • So this is one way to explicitly define 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 more traditional function notation,

  • I could have written a of k, where

  • k is the term that I care about. a

  • of k is equal to 1 plus 3 times k minus 1.

  • This is essentially a function, where

  • an allowable input, the domain, is

  • restricted to positive integers.

  • Now, how would I denote this business right over here?

  • Well, I could say that this is equal to--

  • and people tend to use a.

  • But I could use the notation b sub k or anything else.

  • But I'll do a again-- a sub k.

  • And here, we're going from 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 we wanted to define it explicitly

  • as a function-- we could write this sequence as a sub k, where

  • k starts at the first term and goes to infinity,

  • with a sub k is equaling-- so we're starting at 3.

  • And we are adding 4 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 less than the term that we're at.

  • So it's going to be plus 4 times k minus 1.

  • So this is another way of defining

  • this infinite sequence.

  • Now, in both of these cases, I defined it

  • as an explicit function.

  • So this right over here is explicit.

  • That's not an attractive color.

  • Let me write this in.

  • This is an explicit function.

  • And so you might say, well, what's

  • another way of defining these functions?

  • Well, we can also define it, especially something

  • like an arithmetic sequence, we can also define it recursively.

  • And I want to be clear-- not every sequence can be defined

  • as either an explicit function like this,

  • or as a recursive function.

  • But many can, including this, which

  • is an arithmetic sequence, where we

  • keep adding the same quantity 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 equals 1 and going to 4 with.

  • And when you define a sequence recursively,

  • you want to define what your first term is, with a sub 1

  • equaling 1.

  • You can define every other term in terms of the term before it.

  • And so then we could write a sub k

  • is equal to the previous term.

  • So this is a sub k minus 1.

  • So a given term is equal to the previous term.

  • Let me make it clear-- this is the previous term, plus-- in

  • this case, we're adding 3 every time.

  • Now, how does this make sense?

  • Well, we're defining what a sub 1 is.

  • And if someone says, well, what happens when k equals 2?

  • Well, they're saying, well, it's going to be a sub 2 minus 1.

  • So it's going to be a sub 1 plus 3.

  • Well, we know a sub 1 is 1.

  • So it's going to be 1 plus 3, which is 4.

  • Well, what about a sub 3?

  • Well, it's going to be a sub 2 plus 3. a sub 2,

  • we just calculated as 4.

  • You add 3.

  • It's going to be 7.

  • This is essentially what we mentally

  • did when I first wrote out the sequence, when I said, hey,

  • I'm just going to start with 1.

  • And I'm just going to add 3 for every successive term.

  • So how would we do this one?

  • Well, once again, we could write this as a sub k.

  • Starting at k, the first term, going

  • to infinity with-- our first term, a sub 1,

  • is going to be 3, now.

  • And every successive term, a sub k,

  • is going to be the previous term, a sub k minus 1, plus 4.

  • And once again, you start at 3.

  • And then if you want the second term,

  • it's going to be the first 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 with kind of a base case.

  • And then every term is defined in terms of the term

  • before it or in terms of the function itself,

  • but the function for a different term.

What I want to do in this video is familiarize ourselves