Name: 秦皇島市委宣傳部 - 秦皇島市教育局 (Explicit and recursive definitions of sequences | Precalculus | Khan Academy)
Uploaded: 2021-01-14T06:42:28.000Z
Duration: 8 min 18 s
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反身代名詞

What I want to do in this video is familiarize ourselves

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.

And let's say I only have these four terms right over here.

So this one we would call a finite sequence.

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

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

The sequences where you keep adding the same amount,

which we will also explore in more detail.

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

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

But I want to make us comfortable with how

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

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

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

And this right over here would be the first term.

This right over here would be the second term.

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

I could essentially do it defining our sequence

as explicitly using kind of a function notation or something

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

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

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

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

If k is equal to 1, you're going to get 1 minus 1 is 0.

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 this is one way to explicitly define our sequence with kind

I want to make it clear-- I have essentially

If I wanted a more traditional function notation,

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

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

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

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

And here, we're going from our first term--

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.

For the fourth term, we add 4 three times.

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秦皇島市委宣傳部 - 秦皇島市教育局 (Explicit and recursive definitions of sequences | Precalculus | Khan Academy)

essentially

bunch

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