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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.

• It's going to be 7.

• This is essentially what we mentally

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

• 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