WEBVTT mathematics/math-analysis/selhorst-jones
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Hi--welcome back to Educator.com.
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Today, we are going to talk about an introduction to sequences.
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For the most part, a sequence is simply an ordered list of numbers.
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While the idea is simple, there are a huge variety of uses for sequences.
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They come up in a wide variety of fields, and they are an extremely important tool in advanced mathematics.
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In this lesson, we will learn what a sequence is, various ways to describe them, and how to find patterns that they may be based on.
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Let's go: the definition of a sequence: a sequence, in math, means pretty much the same thing as it does in English.
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It is an order of things; specifically, in this case, it is an ordered list of numbers.
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We could write a sequence as a₁, a₂, a₃, a₄...a<font size="-6">n</font>...
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We call each of the entries in the sequence a term.
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This would be a₁, the first term, because it is the first in the sequence.
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a₂ is the second term, because it is the second in the sequence.
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a₃ would be the third term, because it is the third in the sequence.
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So, any symbol can be used to denote the sequence.
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In this case, we are just the using the lowercase letter a, but we could use any letter, or any symbol, that we wanted.
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a is a common, convenient one, though.
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The subscript (that is the small number to the right: here, this little number 4, a₄) tells us which term it is in the sequence.
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This 4 here tells us that it is the fourth term in the sequence.
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Here are some examples: we could have 1, 2, 3...(and the ... just says "keep going in this manner; it continues on").
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2, 9, 16...; √x, √2x, √3x...a sequence is just some ordered list of things--an ordered list of numbers, in this case.
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So, even here, it ends up being an ordered list of numbers.
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It is using a variable, but once we set x as some number, it is going to end up just being an ordered list of numbers there, as well.
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All right, if a sequence goes on forever without stopping, it is called an infinite sequence.
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Most of the sequences that we are going to work with are infinite sequences.
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That is something where it is a₁, a₂, a₃, a₄...
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and then there is nothing after those dots; it just says that it keeps going, and there is no stop to this thing.
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On the other hand, we could have a finite sequence, if the sequence does stop.
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In that case, we have a₁, a₂, a₃, a₄, and there is that ...
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that says to continue in this manner; but then, we actually stop at a<font size="-6">k</font>.
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Notice how there is nothing after the a<font size="-6">k</font>; it is just blank after that.
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That says that we have reached the endpoint; there is not ... to tell us to keep going in this manner.
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We get to a<font size="-6">k</font>, and we just stop at a<font size="-6">k</font>, because there is nothing after a<font size="-6">k</font>.
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So, it says that this is the end of our sequence.
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We call the number of terms in a finite sequence its **length**.
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In this case, in the length of the above sequence, we would have k, because we have a<font size="-6">k</font> here and we have a₁ here.
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So, that means we are counting our first term, our second term, our third term...all the way up until our kth term.
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1 up until k...if we count from 1 to k, whatever k is, that means we are going to have
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a total of k things, so we have a length of k for that finite sequence.
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We can often talk about some formula that allows us to find the nth term, also called the general term.
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If we know such a formula, we can easily find any term.
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By plugging in different values for n, since we know what the nth term is going to be,
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well, if we say some value for n, we can find the term that is that value.
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If we plug in n = 3, we can find the third locations.
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So, as long as we know that a<font size="-6">n</font> equals some stuff, some algebraic formulation,
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then if we set n equal to 1, we would get the first term, a₁.
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If we set n equal to 17, we get the 17th term, a<font size="-6">17</font>.
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Notice how the n = 17 replaces where the n would have been; it is a subscript n;
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but since we swapped it out for 17, we now have a, subscript 17.
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So, because we have some algebraic formulation for the way a<font size="-6">n</font> works,
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for the way this nth term works, the way this general term works,
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we can plug in our value for n, use this algebraic formulation, and churn out some number to know what a for any term is going to be.
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For example, if we know that a<font size="-6">n</font> = 7n - 5, then we have the sequence a₁, a₂, a₃, a₄...
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Well, notice here: in a₁, that means n = 1; so we swap out the n in 7n - 5 for 7(1) - 5.
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7 - 5 gets us 2, so we now know that a₁ is equal to 2.
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The same thing for a₂: we know that, at a₂, we have n = 2.
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So, we swap out 7n - 5 to 7(2) - 5; 7(2) is 14; 14 - 5 is 9.
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So now, we have that the second entry, the second term, in our sequence is 9.
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a₃: we have n = 3, so we swap out; we have 7(3) - 5, 21 - 5, 16; so our third entry, a₃, is equal to 16.
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At a₄, we have n = 4; 7(4) - 5...7 times 4 is 28; 28 - 5...so we have 23 for our fourth term, as well.
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So, by knowing the general term, a<font size="-6">n</font> = 7n - 5, we are able to find any term.
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We can find any term if we know this general form a<font size="-6">n</font> = some algebraic format, like 7n - 5.
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We can also define a sequence based on terms that came previously.
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We just figured out a way to just say that the absolute thing is going to be this; this will be this, based on this formula.
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We have this definite, general term.
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But we can also define it based on terms that came previously.
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This is called defining a sequence recursively.
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In this, the sequence is built on a recursion formula that shows how each term is based on preceding terms.
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Recursive: we are looking backwards to something that came previously.
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For example, if we have the recursion formula a<font size="-6">n</font> = a<font size="-6">n - 1</font> + 7, what is that saying?
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It is that the nth term--that is a<font size="-6">n</font>, right here--is created by looking at the previous term.
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Well, what would be one before n? n - 1; so a<font size="-6">n - 1</font> is going to be the term just before the nth term.
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So, a<font size="-6">n - 1</font>...and then adding 7 to it is that + 7 business right there.
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a<font size="-6">n</font> = a<font size="-6">n - 1</font> + 7: some term is equal to the previous term, plus 7.
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Since this is true for any n at all, the recursion formula tells us that every term is equal to the term before it, plus 7.
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We didn't say n has to be some specific value; we haven't nailed down what the value of it is going to be.
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So, since this is true for any n, the recursion formula tells us that every term (because it is true for any n,
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so a<font size="-6">n</font> = a<font size="-6">n - 1</font> + 7 for any value of n) will be equal to the term before, plus 7.
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However, there is one special term that doesn't have a term before it.
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Our recursion formula was based on looking at the one behind you and adding 7.
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But in this case, there is one number that isn't going to have anything behind it.
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The person who is the first in line (not really a person--a number)--whatever term is first, our first term: there is nothing behind it.
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There is nothing to look at behind that term.
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So, if that is the case, we need something to start from.
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A recursion formula on its own is not enough to obtain a sequence; we need some sort of starting place before we can make a sequence.
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We need to know what that first term is, what that seed is that our recursion formula will grow off of.
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This is called the initial term (or terms, if we need multiple of them).
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So, using the initial term a₁ = 2, with the previous recursion formula a<font size="-6">n</font> = a<font size="-6">n - 1</font> + 7,
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then our first term is going to be a₁, right here.
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Well, we were just told that a₁ = 2; so that means that we have 2 here.
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Then, from there on, we have a<font size="-6">n</font> = a<font size="-6">n - 1</font> + 7.
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So, that says that to get a term, you take the previous term, and you add 7.
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So, to get from 2 to the next term, we get + 7, so 2 + 7 gets us 9.
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To get to the next one, we have + 7; so that gets us 16.
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To get to the next term, we have + 7; that gets us 23.
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Writing out exactly what happens: if we want to know what a₂ is going to be equal to
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(a₂ is this one right here), then a₂ = a<font size="-6">2 - 1</font>, or 1, + 7.
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So, a₁ is equal to 2; we figured that out here; then + 7...so we get a₂ = 9, which is what we got right here.
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And the same thing is going on for figuring out a₃: a₃ is going to be equal to a₂ + 7.
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a₄ is going to be equal to a₃ + 7, because our recursion formula is telling us to go that way.
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Given a recursion formula and initial terms, it can be possible to find a formula for the nth term.
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That absolute plug-in-a-number-for-our-n...using the general term, it just puts out what the value is.
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There are sometimes ways to be able to do this; if you have a recursion formula and initial terms,
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you can sometimes transform it into a formula for the nth term.
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Similarly, it can be possible to transform an nth term formula into a recursion formula and initial terms.
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So, if we know the general term, we can go to the recursion formula.
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If we go to the recursion formula, we can go to the general form.
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However, we can't always end up doing this.
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There is no guarantee that we can do this; we very often can, especially at this level in math.
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But sometimes it is not going to be so easy to do; sometimes it is going to be really hard.
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Sometimes, it will be totally easy; but sometimes it is going to be hard, and sometimes it is going to be impossible.
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It will depend on the specific sequence that we are working with.
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Some sequences are really easy to talk about in a recursion formula, but basically impossible to talk about as a general term, an nth term format.
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Other ones are really easy to talk about in that nth term format, that general term, but really hard,
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practically impossible, to talk about in a recursion formula.
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So, we won't necessarily be able to switch between the two, but we will often be able to switch between the two.
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And if a problem ever asks us to switch between the two, it will certainly be possible.
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Very, very often, you will be given the first few terms of a sequence and told to either give more terms or figure out a formula for the nth term.
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To do this, you will have to figure out some pattern in the sequence, and then exploit it.
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You will have to look at the pattern, look at the sequence, and ask how these things are connected.
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What is a pattern in here that I can use to create a formula?
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However, before we learn how to do this, before we learn how to find patterns and sequences,
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I want to point out that there is technically no guarantee that a sequence must have a pattern.
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There is no guarantee that we will have patterns in our sequences.
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For example, the below is a perfectly legitimate sequence with no pattern that we are going to be able to find in it.
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47, then -3, then .0012, then π raised to the negative fifth power, then 17, then 1, then 1 again, then 800, and then a whole bunch of other numbers.
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There is no pattern going on here; there is no rhyme; there is no reason.
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There is nothing that we are going to be able to figure out to create some formula here.
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So, there is no guarantee that a sequence has to have some pattern that we are going to be able to create a formula from.
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Things can be really confusing with sequences, and we won't be able to figure out a way to generate a formula.
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But good news: all of the sequences at this level in math will have patterns.
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Technically, a sequence is not required to have a pattern; but at this level in math, all of the sequences that we see are definitely going to have patterns.
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We will always be able to find patterns in the sequences that we are working with.
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So, we don't have to worry about problems being unsolvable, because we can always rely on the fact
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that there is going to be some pattern in there somewhere; there will always be a pattern that we can find, if we look hard enough.
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If the problem is about finding patterns, there is definitely going to be a pattern for us to find.
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We just have to look carefully and be really creative.
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It won't necessarily be easy to find the pattern; but it will be in there somewhere.
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It is not going to be something like this, where it is really, really hard for us to be able to see a pattern,
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because there is simply no pattern; there are going to be cases...
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in all of the things that we are looking at, it is always going to be the case that we are going to be able to find a pattern somehow.
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We don't have to worry about the stuff where there is just no pattern whatsoever.
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All of the problems that we will be asked to do, we will be able to do.
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To find more terms in a sequence, or figure out a formula for the nth term,
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the first thing that we have to do is identify a pattern in the sequence.
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If we want to find a formula, if we want to be able to talk about more terms,
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the first thing that we have to figure out is what goes on in the sequence--how does the sequence work?
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Consider these two sequences: 17, 12, 7, 2... and 2, 6, 18, 54...continuing on with both of them.
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Let's look at the first one: 17, 12, 7, 2: what we can do is say, "Well, what is the connection here between 17 and 12,
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and how is that related to 12, 7; and how is that related to 7 to 2?"
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Well, looking at this for a little while, we probably realize that what we are doing,
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to get from 17 to 12, is subtracting by 5; what we are doing to get from 12 to 7 is subtracting by 5;
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what we are doing to get from 7 to 2 is subtracting by 5.
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So, we know that this pattern of subtract by 5, subtract by 5, subtract by 5...it is going to continue on,
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because it showed up everywhere in the sequence so far.
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Similarly, over here, how do we get from 2 to 6, from 6 to 18, from 18 to 54?
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Looking at it for a while, we probably realize that what we are doing is multiplying by 3.
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2 times 3 gets us 6; 6 times 3 gets us 18; 18 times 3 gets us 54; so this pattern will continue on,
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throughout the rest of the sequence, because it showed up in all of the sequence that we saw.
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We have figured out what the patterns are.
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We notice that the sequence on the left subtracts by 5 every term; the one on the right multiplies by 3.
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At this point, it would be easy to find more terms.
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If we wanted to find the next term in the sequence 17, 12, 7, 2, we would just subtract 5 again; 2 - 5 would get us -3.
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And we would be able to keep going, if we wanted to.
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Similarly, over here, if we wanted to figure out the next term in 2, 6, 18, 54, we would just have to multiply by 3 again.
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So, what would come after that? 54 times 2 is 150 + 4(3) is 12, so 162.
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So, we would get 162 as the next one, and we would be able to continue on in that manner, if we wanted to find any more terms in the sequence.
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It is easy to find more terms, and it would also be easy to find a recursion formula.
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The red one that we were doing, the minus 5 one, is just going to be a<font size="-6">n - 1</font> - 5.
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And the green one, the second one, with multiplication, times 3...that would be a<font size="-6">n - 1</font> times 3.
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So, it wouldn't be very hard to figure out recursion formulas, because they are really deeply connected to the pattern we saw.
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However, if we want a formula for the nth term, it is going to take a little more thought,
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because we have to be able to figure out how this works for all of them.
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It is not just describing the pattern of how we get from one to the next, or how we get from this one to the next one.
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It is describing how we get to any of them, without being able to have any sort of reference points.
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We can't just say - 5, - 5, - 5; we have to figure out a way of collecting all of the - 5s that happened, or all of the times 3s that happened.
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So, we think about it for a while; and we will be able to figure this out.
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We would be able to get a<font size="-6">n</font> = 22 - 5n; we would be able to realize that,
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since what we are doing is subtracting by 5 a bunch of times, it is going to be - 5 times n;
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and then we need to figure out what number we are subtracting 5 from.
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And we want to make sure to check the first few terms.
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Always check the first few terms, once you think you have figured out a formula for the general term.
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Once you think you know what the nth term is going to be, make sure you check that what you figured out is right,
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because it is easy to make a mistake and be off by a little bit, to be off by one number in the sequence.
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So, just make sure that you try and check.
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For example, if we have figured out what a₁ is going to be, well, that would be 22 - 5(1).
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22 - 5(1) is 22 - 5, or 17; that checks out with what we have here.
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If we did a₂, then that would be equal to 22 - 5(2); 22 - 5(2) is 22 - 10; 22 - 10 is 12; that checks out with what we have here.
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So, it looks like it ends up working out.
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Similarly, for the green one, our multiplication one, a<font size="-6">n</font> = 2(3)^n - 1:
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well, let's think about this--does this end up working out?
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We realize that it has to be something about 3 to the some exponent, because every step, we are multiplying by some 3.
00:16:48.900 --> 00:16:54.400
So, if we stack all of those 3s together, it is going to be 3 to the some sort of exponent.
00:16:54.400 --> 00:17:01.300
The question is what that exponent should be: it is 3^n - 1, because this very first one hasn't been affected by the 3 at all.
00:17:01.300 --> 00:17:03.100
Let's check and make sure that that is the case.
00:17:03.100 --> 00:17:11.700
If we plug in a₁ = 2(3)^1 - 1, then that would be 2(3)⁰; 3⁰ is just 1;
00:17:11.700 --> 00:17:16.600
any number raised to the 0 is just 1; so we get 2; 2 checks out.
00:17:16.600 --> 00:17:25.300
If we did a₂, then we would have a₂ is 2(3)^2 - 1, or to the 1; so it equals 6; that checks out.
00:17:25.300 --> 00:17:29.300
We can see that this is going to continue to work; so our general formula works out.
00:17:29.300 --> 00:17:32.200
But make sure you check it and think about it; it can be a little bit difficult,
00:17:32.200 --> 00:17:36.400
but as long as you check it, you can be sure that what you have is going to work.
00:17:36.400 --> 00:17:42.000
When trying to recognize a pattern in a sequence, try to think in terms of how to get from one term to the next.
00:17:42.000 --> 00:17:46.200
How do you get from this first term to the next term?
00:17:46.200 --> 00:17:48.200
How do you get from the first term to the second term?
00:17:48.200 --> 00:17:54.100
Establish a hypothesis, some guess at what you think the pattern is, by looking a₁ to a₂.
00:17:54.100 --> 00:17:58.300
You want to start with a hypothesis--you think that this is probably how the pattern works.
00:17:58.300 --> 00:18:03.200
You look at a₁ to a₂, and then you want to test that hypothesis against the following ones:
00:18:03.200 --> 00:18:07.000
a₂ to a₃, a₃ to a₄, and any other terms that are given.
00:18:07.000 --> 00:18:11.000
You come up with thinking that the pattern here, the way we get from one stepping-stone
00:18:11.000 --> 00:18:13.800
to the next stepping-stone, is that we do some operation.
00:18:13.800 --> 00:18:18.800
Add some number, multiply by some number...it is doing some sort of thing; how is it working out?
00:18:18.800 --> 00:18:21.600
Figure out what you think is going on for the way that the pattern works.
00:18:21.600 --> 00:18:25.900
And then, test and make sure that that works on the way that we get to the next one and the way that we get to the next one,
00:18:25.900 --> 00:18:29.400
or that it just works for the nth number location.
00:18:29.400 --> 00:18:32.100
Once you have some hypothesis, test it against all of the other ones.
00:18:32.100 --> 00:18:36.400
If it works, great; you just figured out the pattern--now you are ready to figure out some way
00:18:36.400 --> 00:18:39.800
to formulate that general form, that general term, the nth term.
00:18:39.800 --> 00:18:48.300
If it doesn't work, go back to the beginning: figure out a new hypothesis, and then try again.
00:18:48.300 --> 00:18:54.100
It is all about figuring out something that you think might work, and then testing it against the information you have.
00:18:54.100 --> 00:18:57.100
Keep testing until you get something that actually ends up working out,
00:18:57.100 --> 00:19:03.200
at which point, you have found the pattern; now you are ready to start working your way towards a general term formula.
00:19:03.200 --> 00:19:06.600
Once you figure out the pattern, it is easy to find further terms in the sequence.
00:19:06.600 --> 00:19:12.700
That is the easiest part; you just continue with that pattern to generate any more terms that they tell you to generate.
00:19:12.700 --> 00:19:19.500
Finding a formula for the nth term: that could be tricky--a formula for the nth term can be a little bit tricky.
00:19:19.500 --> 00:19:23.700
What you want to do is think carefully about how you can put the pattern into an equation
00:19:23.700 --> 00:19:26.200
and make sure to check some terms after you create the formula.
00:19:26.200 --> 00:19:33.000
That checking is really, really important; think carefully about how you can put that pattern into an equation,
00:19:33.000 --> 00:19:37.600
and then check after you have come up with some sort of equation that you think will probably work.
00:19:37.600 --> 00:19:41.700
It is really important to check, because it is really easy to make mistakes, especially the first couple of times you are doing it.
00:19:41.700 --> 00:19:43.900
We will also see this a whole bunch of times in the examples.
00:19:43.900 --> 00:19:45.900
We are going to work with a whole bunch of examples here.
00:19:45.900 --> 00:19:48.500
So, that will really help to cement our understanding of how to do this.
00:19:48.500 --> 00:19:51.400
We have lots of examples to make this clear.
00:19:51.400 --> 00:19:55.200
All right, how do we find patterns?--that can sometimes be a tricky thing.
00:19:55.200 --> 00:20:02.200
When trying to find the pattern in a sequence, the two most common pattern types that appear are addition/subtraction,
00:20:02.200 --> 00:20:07.100
where we just add some k every term (k could be a positive number; k could be a negative number;
00:20:07.100 --> 00:20:13.600
that allows us to add or subtract, depending on what the k is; but we are just adding the same k, adding some constant number,
00:20:13.600 --> 00:20:20.700
every time we do a step); and then the other one is multiplication/division, where we multiply by some k every term.
00:20:20.700 --> 00:20:25.400
If it is multiplication, it is just some constant number k; and we can also effectively divide.
00:20:25.400 --> 00:20:29.300
If it is a fraction as our k, we are effectively doing division there, as well.
00:20:29.300 --> 00:20:31.500
So, we are just multiplying by some constant number every term.
00:20:31.500 --> 00:20:34.900
Every step is either going to be addition or multiplication.
00:20:34.900 --> 00:20:40.100
This is the largest portion of the patterns; many, many of the patterns that we are going to work with,
00:20:40.100 --> 00:20:45.400
at this level, and really at any level, are going to be connected to addition/subtraction or multiplication and division.
00:20:45.400 --> 00:20:48.000
So, these are the first two that you want to keep in your head.
00:20:48.000 --> 00:20:53.700
A large number of patterns can be figured out just by keeping these two types in mind.
00:20:53.700 --> 00:20:57.100
Always think first in terms of addition/subtraction, multiplication/division.
00:20:57.100 --> 00:20:59.600
Check to see if you see those first.
00:20:59.600 --> 00:21:02.500
However, those are not the only kinds of patterns that you end up seeing.
00:21:02.500 --> 00:21:06.700
These two pattern types are not enough to figure out the pattern for all sequences.
00:21:06.700 --> 00:21:10.000
In that case, it can help to keep various other patterns in mind.
00:21:10.000 --> 00:21:18.200
A good one to keep in mind is the squares: n²: 1², 2², 3², 4², 5², 6², 7²...
00:21:18.200 --> 00:21:23.000
But often, you won't see them as a number squared, because then it would be really easy to recognize the pattern.
00:21:23.000 --> 00:21:31.500
Instead, you see it as 1, 4, 9, 16, 25, 36, 49...and continuing on, if they still have even more terms.
00:21:31.500 --> 00:21:35.100
So, it just helps to keep that structure of numbers in the back of your head.
00:21:35.100 --> 00:21:39.400
It is really useful to be able to remember what all of those perfect squares are, what all those squares are
00:21:39.400 --> 00:21:43.500
that you are used to working with, that you have seen in previous algebra classes.
00:21:43.500 --> 00:21:48.900
Just keep them in your mind, and look for things that look somewhat like that, or along those lines.
00:21:48.900 --> 00:21:54.300
Another one that often shows up is cubes; this one is less often than squares, but it does show up.
00:21:54.300 --> 00:21:59.100
n³ is 1³, 2³, 3³, 4³, 5³, 6³, etc.
00:21:59.100 --> 00:22:04.500
Once again, you very often won't see it as a number cubed, because then it would be easy to see that the pattern is cubed.
00:22:04.500 --> 00:22:06.500
And it is supposed to be a little more challenging than that.
00:22:06.500 --> 00:22:14.100
So instead, it is normally 1, and then 8, and then 27, and 64, then 125, and 216, and so on, and so on.
00:22:14.100 --> 00:22:20.300
So, you are probably less used to using cubes; it is really important to just pay attention to at least these first four.
00:22:20.300 --> 00:22:27.800
Memorize 1, 8, 27, and maybe 64, maybe 125; keep at least those first few terms in mind,
00:22:27.800 --> 00:22:33.400
because you want to be prepared to say, "Well, I am not used to seeing this pattern; I am not used to seeing something like this;
00:22:33.400 --> 00:22:38.300
but oh, maybe it is cubes, because I see that the first three are like that," and then you can check the other ones
00:22:38.300 --> 00:22:40.500
by hand, and make sure that that does work out.
00:22:40.500 --> 00:22:42.800
You don't have to memorize the whole thing, but you do have to be ready,
00:22:42.800 --> 00:22:46.500
when you see the pattern, to be able to think that maybe that is going to be something.
00:22:46.500 --> 00:22:50.700
You have to be prepared to recognize it; you don't have to know the whole pattern, but you have to be prepared to recognize it.
00:22:50.700 --> 00:23:02.700
Finally, factorials: n!: 1!, 2!, 3!, 4!, 5!, 6!, etc., etc.: once again, you aren't normally going to see that as factorials written out.
00:23:02.700 --> 00:23:13.500
You will instead often see it as something like 1, then 2, then 6, 24, 120, 720...a really good one to memorize is 1, 2, 6, 24, 120.
00:23:13.500 --> 00:23:21.900
That is 1!, then 2!, 3!, 4!, 5!; so if you can keep that pattern in your head--just keep that one in the back of your head--
00:23:21.900 --> 00:23:25.600
then that will end up showing up a lot, and if you are not prepared to recognize that pattern,
00:23:25.600 --> 00:23:30.100
that kind of problem would be really, really hard, because you won't be able to see that pattern when it shows up.
00:23:30.100 --> 00:23:39.000
And in case you forgot how a factorial works, factorials multiply the number that is factorial by every integer below the number.
00:23:39.000 --> 00:23:46.200
So, for example, 5! would be 5 times 4 times 3 times 2 times 1, which works out to 120.
00:23:46.200 --> 00:23:51.400
And we define...we just specifically define 0! = 1 for ease.
00:23:51.400 --> 00:23:54.600
It helps things out; it helps a lot of other things in math work out.
00:23:54.600 --> 00:24:02.700
So, 0! = 1; we just set it that way and try not to get worried about how that doesn't make sense, compared to how the other one works out.
00:24:02.700 --> 00:24:08.700
It actually sort of makes sense; but it is better to just remember and memorize 0! = 1.
00:24:08.700 --> 00:24:10.900
That one will come up occasionally.
00:24:10.900 --> 00:24:18.300
All right, sometimes the sign will change with each term; it will flip between positive and negative, positive/negative, positive/negative.
00:24:18.300 --> 00:24:23.400
You will see a positive on one, and the next one will be negative, and the next one will be positive, and the next will be negative.
00:24:23.400 --> 00:24:26.100
And the next one will be positive, and then negative, and so on, and so forth.
00:24:26.100 --> 00:24:34.100
And you will see this flipping pattern; if that is the case, it might be one of these two following--one of these two methods, these two types of patterns.
00:24:34.100 --> 00:24:42.400
-1 raised to the n + 1: notice that, if we have n at 1, then we will have -1 squared, which is going to come out to be a positive 1.
00:24:42.400 --> 00:24:49.200
And then, the next one would be -1 to the 2 + 1; n = 2, so -1 to the 3 is going to end up being -1.
00:24:49.200 --> 00:24:56.400
And then, we get +1 and -1 and +1 and -1, because -1 raised to some integer is just going to multiply by -1 that many times.
00:24:56.400 --> 00:25:01.400
So, it will flip between positive and negative, positive/negative, as our n's step up, one at a time.
00:25:01.400 --> 00:25:06.200
Similarly, -1 raised to the n gives us the exact same pattern, flipping and turning.
00:25:06.200 --> 00:25:13.600
It just starts, instead of starting at +1, at -1; and then, it will be +1 and -1, then +1, then -1, then +1.
00:25:13.600 --> 00:25:19.700
So, if you see a flipping sign pattern, and you don't see something else to be able to cause that to happen,
00:25:19.700 --> 00:25:23.200
in the sequence that you are working with, whatever pattern you are working with,
00:25:23.200 --> 00:25:32.100
these two right here are a really good thing to keep in the back of your mind for a way to just cause a flipping sign to appear.
00:25:32.100 --> 00:25:35.600
If most of the terms in the sequence are presented in a certain format--for example,
00:25:35.600 --> 00:25:39.600
they are all in fractions--try to figure out a way to put all of the terms in that format.
00:25:39.600 --> 00:25:45.000
So, if you see a certain format in your sequence, put all of the terms into that format.
00:25:45.000 --> 00:25:49.400
If most of your terms are in fractions, make all of your terms in fractions.
00:25:49.400 --> 00:25:54.700
It can be a lot easier to see patterns if everything is in the same format.
00:25:54.700 --> 00:26:01.700
So, if you get all of your sequence terms in the same format, it normally causes patterns to appear more readily.
00:26:01.700 --> 00:26:05.500
Furthermore, if the format can clearly be broken into multiple parts--for example,
00:26:05.500 --> 00:26:14.700
if we have a fraction, _/_, we can break it into the numerator (the part on top) and the denominator (the part on the bottom)--
00:26:14.700 --> 00:26:17.600
in that case, we can clearly talk about how all of our numerators
00:26:17.600 --> 00:26:21.200
behave by some pattern, and all of our denominators behave by some pattern.
00:26:21.200 --> 00:26:27.500
If you can break the thing into multiple parts, if a format can be broken into multiple parts--
00:26:27.500 --> 00:26:35.700
for example, a fraction is a numerator over a denominator, every single time--it can help to figure out patterns for each part separately.
00:26:35.700 --> 00:26:39.200
Figure out the numerator pattern on its own; figure out the denominator pattern on its own.
00:26:39.200 --> 00:26:43.300
Sometimes, that will really help clarify things; you don't have to worry about trying to figure out the whole fraction.
00:26:43.300 --> 00:26:49.400
You can instead break it apart piecemeal and then just put them back together once you recognize each pattern on its own.
00:26:49.400 --> 00:26:59.200
It is important to note that all of these different pattern types that we have talked about so far-- they don't necessarily occur in isolation.
00:26:59.200 --> 00:27:03.800
While that will sometimes happen (you will sometimes just have addition; you will sometimes just have multiplication;
00:27:03.800 --> 00:27:08.800
you will sometimes just have switching signs), we are often going to end up working with sequences
00:27:08.800 --> 00:27:16.800
that use multiple pattern types at once, so it will be up to us to figure out that it is using this pattern and this pattern and this pattern,
00:27:16.800 --> 00:27:21.800
and then figure out some way to merge all of those three patterns together, once we are trying to describe the whole thing.
00:27:21.800 --> 00:27:25.300
They might even end up involving patterns that you haven't seen before.
00:27:25.300 --> 00:27:29.900
They might do something that you don't immediately recognize; we have to come up with a new way to describe it.
00:27:29.900 --> 00:27:34.300
So, keep a lookout for something that looks weird, that is totally new to the way that you are doing things.
00:27:34.300 --> 00:27:40.300
And you might end up having to learn a new method of describing things, which will just take longer to think through.
00:27:40.300 --> 00:27:44.200
It can sometimes help to write the number of the term above or below each term,
00:27:44.200 --> 00:27:48.700
to write n = 1 and then n = 2, and so on and so forth.
00:27:48.700 --> 00:27:55.800
By writing the number of the term above or below, you are able to keep track of numerical location.
00:27:55.800 --> 00:28:00.200
By being able to see what number we are at--are we at the first term? Are we at the fifth term?--
00:28:00.200 --> 00:28:04.700
by being able to have this clear reference point of "this is term #1; this is term #5,"
00:28:04.700 --> 00:28:07.700
you will be able to see how the number of the term relates
00:28:07.700 --> 00:28:14.000
to the values inside of the actual term inside of the sequence--the values of that term.
00:28:14.000 --> 00:28:18.100
The location of the term will normally be related to the values inside of the term.
00:28:18.100 --> 00:28:23.800
That will often make it easier to identify patterns, by being able to see that this has number location 5;
00:28:23.800 --> 00:28:27.900
and because of that, we see that the general form is working in this certain way.
00:28:27.900 --> 00:28:32.400
So, writing the numbers above or below can really help you see that sort of thing.
00:28:32.400 --> 00:28:36.100
In the end, there is no one way to identify all patterns.
00:28:36.100 --> 00:28:43.000
I want you to try to take a broad view of the sequence and look for repetitions or similarities to other patterns that you have seen.
00:28:43.000 --> 00:28:46.600
Don't try to focus on it always being the same thing, because it won't.
00:28:46.600 --> 00:28:49.200
Each sequence is probably going to have its own special pattern.
00:28:49.200 --> 00:28:53.600
You will start getting used to certain ways that patterns interact, or certain types of patterns.
00:28:53.600 --> 00:28:56.200
And it will be easier to recognize them on future ones.
00:28:56.200 --> 00:29:01.100
And a lot of this stuff will show up in a whole bunch of different places, like standardized tests,
00:29:01.100 --> 00:29:08.500
later in different math classes, in science classes...so what you are learning in this class will definitely be applicable for a long time to come.
00:29:08.500 --> 00:29:10.900
But it is not necessarily always going to be the same thing.
00:29:10.900 --> 00:29:14.100
So, just take a broad view of what is going on.
00:29:14.100 --> 00:29:19.700
Don't think that it is definitely going to work in one way, because you don't really know until you have figured out what the pattern is.
00:29:19.700 --> 00:29:25.900
So, look at the thing carefully; think, "How does one term interact with the next term?"
00:29:25.900 --> 00:29:28.200
How are these two terms related to each other?
00:29:28.200 --> 00:29:34.600
Is there some sort of general pattern that is occurring as I look through all of the numbers, all of my terms, at once?
00:29:34.600 --> 00:29:39.900
Try to think in really large, broad strokes before you try to come up with a very specific pattern showing up.
00:29:39.900 --> 00:29:44.200
And if you still can't figure it out, if you are looking at it for a long time and you can't figure it out,
00:29:44.200 --> 00:29:47.100
see if there is an alternative way to write the terms out.
00:29:47.100 --> 00:29:51.300
That was what I was talking about with...if most of them are in fractions, put all of them in fractions.
00:29:51.300 --> 00:29:55.800
Figure out if there is some way to write the terms in something, so that they all have this new alternative way,
00:29:55.800 --> 00:29:58.700
because maybe that will help you see what is going on better.
00:29:58.700 --> 00:30:01.800
And just in general, persevere; be creative.
00:30:01.800 --> 00:30:06.700
Figuring out patterns is not something that is just a step-by-step method, and you have gotten to the answer.
00:30:06.700 --> 00:30:09.300
It is something where you have to look at it and sort of think for a while.
00:30:09.300 --> 00:30:12.400
Just be clever; have a little bit of luck; and just work at it.
00:30:12.400 --> 00:30:17.300
If you really can't figure it out for a long time, go to the next problem and come back to that problem later.
00:30:17.300 --> 00:30:21.100
Sometimes just a little bit of time will cause it to "bounce around" in your head,
00:30:21.100 --> 00:30:24.700
and you will be able to see the pattern easily, when earlier it was really, really difficult.
00:30:24.700 --> 00:30:29.300
So, just stick with it, and as you work with it more and more, it will make more and more sense.
00:30:29.300 --> 00:30:31.500
All right, we are ready for some examples.
00:30:31.500 --> 00:30:35.700
Given the nth term, write the first four terms of each sequence.
00:30:35.700 --> 00:30:38.700
Assume that each sequence starts at n = 1.
00:30:38.700 --> 00:30:43.400
So, that nth term, that general term, is a<font size="-6">n</font> = stuff.
00:30:43.400 --> 00:30:48.900
In this case, for our first one, we have a<font size="-6">n</font> = 3n - 2.
00:30:48.900 --> 00:30:55.200
Our first term, the a₁, is going to be when n = 1.
00:30:55.200 --> 00:31:00.400
We plug in a₁ = 3(1) - 2.
00:31:00.400 --> 00:31:07.400
Similarly, a₂ is when n = 2, so we have a₂ = 3(2) - 2.
00:31:07.400 --> 00:31:20.000
a₃ = 3(3) - 2; a₄ = (I'll write that a little below; there is not quite enough room) 3(4) - 2.
00:31:20.000 --> 00:31:26.400
We work this out: a₁ = 3(1) - 2, 3 - 2, so we get a₁ = 1.
00:31:26.400 --> 00:31:36.100
a₂: 3(2) is 6; 6 - 2 is 4; a₃ = 3(3), 9, minus 2; 9 - 2 is 7.
00:31:36.100 --> 00:31:42.200
a₄ = 3(4) is 12, minus 2 is 10.
00:31:42.200 --> 00:31:49.200
So, we have the first four terms here: a₁ = 1, a₂ = 4, a₃ = 7, a₄ = 10.
00:31:49.200 --> 00:31:52.800
Also, I want to point out; notice how a₁, to get to a₂...we added 3.
00:31:52.800 --> 00:31:56.400
To get to a₃, we added 3; to get to a₄, we added 3.
00:31:56.400 --> 00:32:02.000
Each term here has this + 3 step, which we are going to end up seeing from this 3 times n,
00:32:02.000 --> 00:32:07.700
because the 3 times n...the n is what term location we are at, so as we go up more term locations,
00:32:07.700 --> 00:32:15.300
we are going to end up seeing more times that we have ended up adding on this number 3 to the thing.
00:32:15.300 --> 00:32:27.200
All right, the next one: b<font size="-6">n</font> = -1^n/(n + 3); our first one, a₁, is going to be (-1)¹/(1 + 3).
00:32:27.200 --> 00:32:32.200
We swap out all of the n's that occur for whatever we have here, a₁.
00:32:32.200 --> 00:32:39.600
Next, a₂ = -1 squared, -1 to the 2, divided by 2, plus 3.
00:32:39.600 --> 00:32:45.700
a₃ is equal to -1 to the 3, over 3 + 3.
00:32:45.700 --> 00:32:53.100
a₄ is going to be equal to -1 to the 4, over 4 + 3.
00:32:53.100 --> 00:33:03.700
We can work this out here: we have a₁ = (-1)¹, which is still just -1; 1 + 3 is 4, so we have -1/4.
00:33:03.700 --> 00:33:12.600
a₂: (-1)², -1 times -1...that cancels to just positive, so we just have a positive 1, divided by 2 + 3, 5.
00:33:12.600 --> 00:33:18.400
a₃ = (-1)³; -1 to an odd exponent is going to end up leaving a negative after.
00:33:18.400 --> 00:33:22.100
So, we have -1 over 3 + 3, 6.
00:33:22.100 --> 00:33:27.700
a₄ = (-1)⁴, to an even exponent; it is going to cancel out; we are going to have a positive.
00:33:27.700 --> 00:33:40.000
So, we have 1/(4 + 3) is 7: so a₁ = -1/4; a₂ = +1/5; a₃ = -1/6; a₄ = 1/7.
00:33:40.000 --> 00:33:41.900
We found the first four terms.
00:33:41.900 --> 00:33:50.600
Finally, c<font size="-6">n</font> = 47...oh, oops; that whole time shouldn't have been a<font size="-6">n</font>, because it was b<font size="-6">n</font>.
00:33:50.600 --> 00:33:56.300
So, it actually should have been not a₂, not a₁...any of these...
00:33:56.300 --> 00:34:03.000
It should have been b₁, b₂, b₃, b₄, because it has a different name than the sequence at the top.
00:34:03.000 --> 00:34:07.700
That is why we are using a different letter--because it is a different sequence for this problem.
00:34:07.700 --> 00:34:15.000
b₁, b₂, b₃, b₄...it is easy to end up forgetting that we are changing symbols sometimes.
00:34:15.000 --> 00:34:17.900
So, pay attention to the symbol of the sequence that you are working with.
00:34:17.900 --> 00:34:25.200
All right, the last one: c<font size="-6">n</font> = 47; the thing to notice here is...does this side, the right side, end up involving n at all?
00:34:25.200 --> 00:34:29.300
It doesn't; as the n changes, the right side doesn't notice the n change.
00:34:29.300 --> 00:34:34.600
So, c₁ is going to be equal to 47; c₂ is going to be equal to 47;
00:34:34.600 --> 00:34:40.000
c₃ is going to be equal to 47; c₄ is going to be equal to 47.
00:34:40.000 --> 00:34:44.400
So, whatever value we end up using for n, it is always going to end up coming out to 47.
00:34:44.400 --> 00:34:49.000
So, the first term is 47; the second term is 47; the third term is 47; the fourth term is 47.
00:34:49.000 --> 00:34:53.700
We always end up getting 47, because it is just a constant sequence.
00:34:53.700 --> 00:35:00.100
All right, the second example: the Fibonacci sequence is a well-known, recursively-defined sequence.
00:35:00.100 --> 00:35:05.000
It is given by the recursion formula and the initial terms below; write out the first 12 terms.
00:35:05.000 --> 00:35:14.500
Its recursion formula is a<font size="-6">n</font> = a<font size="-6">n - 1</font> + a<font size="-6">n - 2</font>, and a₁ = a₂, which equals 1.
00:35:14.500 --> 00:35:25.100
Right away, we know that our first two terms are 1 (a₁ = 1), and then a₂ also equals 1; so 1, 1.
00:35:25.100 --> 00:35:30.000
If we want to figure out the next term, we have a<font size="-6">n</font> = a<font size="-6">n - 1</font> + a<font size="-6">n - 2</font>.
00:35:30.000 --> 00:35:35.100
So, if we want to figure out what a₃ is going to be, then that is going to be equal to a<font size="-6">3 - 1</font>,
00:35:35.100 --> 00:35:40.100
so a₂, plus a<font size="-6">3 - 2</font>, a₁.
00:35:40.100 --> 00:35:43.800
a₃...we don't know what a₃ is, but we do know what a₁ and a₂ are.
00:35:43.800 --> 00:35:49.700
They are both 1; so we have 1 + 1; a₃ = 2.
00:35:49.700 --> 00:35:54.900
So, our next term is 2; what comes after that?
00:35:54.900 --> 00:36:04.400
If we want to figure out a₄, that would be a<font size="-6">4 - 1</font>, a₃, plus a<font size="-6">4 - 2</font>, a₂.
00:36:04.400 --> 00:36:15.200
What is a₃? We just figured out a₃ = 2, so we have 2 +...a₂, once again, is 1; that equals 3.
00:36:15.200 --> 00:36:21.200
a₄ = 3, so the next thing is going to be a 3.
00:36:21.200 --> 00:36:25.900
Let's do one more of these, and then we will see what the general pattern here, going on, is.
00:36:25.900 --> 00:36:30.700
a₅ is equal to...not the general term, but how this pattern is working, on the whole...
00:36:30.700 --> 00:36:38.800
a₅ is equal to a₄, the previous term, plus the term previous to that one, a₃.
00:36:38.800 --> 00:36:46.400
So, a₅ = a₄ + a₃: we just figured out that a₄ = 3 and a₃ = 2.
00:36:46.400 --> 00:36:53.000
So, we have a₅ = 5; there is our next term, 5.
00:36:53.000 --> 00:36:54.700
What comes after that--how do we get this?
00:36:54.700 --> 00:37:01.200
Well, notice: a₅ = a₄ + a₃; so a₅ is equal to the previous term, plus the term previous to that.
00:37:01.200 --> 00:37:06.800
a₄ equaled a₃ + a₂; a₄ is equal to the previous term, plus the term previous to that.
00:37:06.800 --> 00:37:11.900
a₃ = a₂ + a₁, so the previous term, plus the term previous to that.
00:37:11.900 --> 00:37:16.900
What we are doing to make the next term: it is looking at the previous term and the previous previous term.
00:37:16.900 --> 00:37:22.900
To make whatever comes after the 5, it is going to be: add 3 and 5 together, and that will make the next thing.
00:37:22.900 --> 00:37:31.600
So, 3 + 5 gets us 8; then, the next one is going to, once again, be: take the 5 and the 8; add them together.
00:37:31.600 --> 00:37:43.100
5 + 8 gets us 13; we see that this relationship here is that any term is equal to the previous term and the previous previous term, added together.
00:37:43.100 --> 00:37:47.200
So, that is why we needed a₁ and a₂, because we needed 2 terms,
00:37:47.200 --> 00:37:49.500
so that we could talk in terms of the previous previous term.
00:37:49.500 --> 00:37:54.000
We needed that larger start; we needed two initial terms before we would be able to get started.
00:37:54.000 --> 00:38:14.000
So, at this point, we just add things together: 8 + 13 is 21; 21 + 13 is 34; 21 + 34 is 55; 34 + 55 is 89; 55 + 89 is 144.
00:38:14.000 --> 00:38:16.600
And it will continue on in this manner.
00:38:16.600 --> 00:38:23.700
And there we are; there is the Fibonacci sequence; there are the first 12 terms of the Fibonacci sequence.
00:38:23.700 --> 00:38:26.300
Cool; the Fibonacci sequence has some other interesting properties.
00:38:26.300 --> 00:38:30.400
You might end up studying it more in the class that you are currently in, or in a future class.
00:38:30.400 --> 00:38:35.600
It is a pretty cool thing; but it is enough for us now just to understand how the thing works out.
00:38:35.600 --> 00:38:41.800
All right, the third example: Find the nth term for each sequence below, and assume that the sequence starts at n = 1.
00:38:41.800 --> 00:38:46.100
The first thing that we always have to do, if we are looking at a sequence, and we want to figure out what the nth term,
00:38:46.100 --> 00:38:55.400
the a<font size="-6">n</font>, the general term, is (it equals some formulaic algebra expression), then it is going to be...
00:38:55.400 --> 00:38:58.600
we have to figure out the pattern, and then we use that pattern to come up with an equation.
00:38:58.600 --> 00:39:04.500
So, what is the pattern in this first sequence? -3, 1, 5, 9, 13.
00:39:04.500 --> 00:39:08.600
Well, notice: how do we get from -3 to 1? We can just add 4.
00:39:08.600 --> 00:39:13.500
How do we get from 1 to 5? We add 4 again--it looks like our pattern probably is going to work out.
00:39:13.500 --> 00:39:21.600
5 to 9--we add 4; 9 to 13--we add 4; 13 to whatever is next--it is probably going to be add 4, add 4, add 4.
00:39:21.600 --> 00:39:24.900
Since the pattern worked for everything that we have seen so far in the sequence,
00:39:24.900 --> 00:39:28.700
we can assume that the pattern is definitely just "add 4."
00:39:28.700 --> 00:39:37.000
So, if that is going to be the case, we know that it has to be something of the form a<font size="-6">n</font> = 4n +...we don't know yet.
00:39:37.000 --> 00:39:39.400
We don't know what it is going to be, so we will just leave it as a question mark.
00:39:39.400 --> 00:39:44.100
We could also use a variable like normal, but in this case, the variable I have decided to use is ?.
00:39:44.100 --> 00:39:51.600
a<font size="-6">n</font> = 4n...that represents this step, step, step, step: +4, +4, +4, +4.
00:39:51.600 --> 00:39:55.900
Every step brings +4; every term you move on to brings this adding by 4.
00:39:55.900 --> 00:40:03.000
And so, 4 times n will allow us to represent how many steps we have taken, multiplied by 4; and that brings that many 4s to the table.
00:40:03.000 --> 00:40:04.800
So now, we just need to figure out what the question mark is.
00:40:04.800 --> 00:40:14.600
Well, notice: we know that a₁ = -3; so that means we can have a₁ = 4(1) + ?.
00:40:14.600 --> 00:40:25.700
We know that a₁ equals -3; so -3 = 4 (4 times 1 is just 4) + ?; so we have -7 (solving for question mark) = ?.
00:40:25.700 --> 00:40:35.800
So, at this point, plugging that in, we have that a<font size="-6">n</font>, the general term, is equal to 4n - 7.
00:40:35.800 --> 00:40:40.500
It is always a good idea to check this sort of thing out; that probably will end up working out, but let's make sure.
00:40:40.500 --> 00:40:42.600
We already checked a₁; that is how we figured it out.
00:40:42.600 --> 00:40:44.800
Let's check and make sure that a₂ ends up working out.
00:40:44.800 --> 00:40:52.100
a₂ = 4(2) - 7, 8 - 7, which equals 1; and that checks out.
00:40:52.100 --> 00:41:00.600
Next, a₃ = 4(3) - 7 = 12 - 7 = 5; and that checks out.
00:41:00.600 --> 00:41:03.800
And we can see, going along, this method--that this is going to end up working--
00:41:03.800 --> 00:41:10.600
because we have this 4n here, so every time we go forward another term, we are going to end up adding another 4,
00:41:10.600 --> 00:41:15.900
which also represents our pattern of + 4 each time; so it makes sense--we have our answer.
00:41:15.900 --> 00:41:20.100
The general term for that sequence is a<font size="-6">n</font> = 4n - 7.
00:41:20.100 --> 00:41:25.200
All right, the next one: 2, 5, 10, 17, 26.
00:41:25.200 --> 00:41:27.900
The first thing we want to do is figure out what the pattern is.
00:41:27.900 --> 00:41:32.000
If it is addition, then we would have something like + 3.
00:41:32.000 --> 00:41:42.500
All right, to get from 2 to 5, it is + 3; to get from 5 to 10, it is + 5; to get from 10 to 17, it is + 7...that is not going to end up working out.
00:41:42.500 --> 00:41:45.700
It can't be addition as our pattern; we see that pretty quickly.
00:41:45.700 --> 00:41:48.600
So, let's try another really common one, which is multiplying.
00:41:48.600 --> 00:41:56.800
Well, to get from 2 to 5, you have to multiply by 5/2; to get from 5 to 10, we multiply by...that is not going to work.
00:41:56.800 --> 00:42:02.200
We see very quickly that multiplication is just not friendly here; it is not going to work out in a very good way.
00:42:02.200 --> 00:42:11.500
So, multiplication is out; at that point, we see that addition fails; multiplication fails.
00:42:11.500 --> 00:42:15.700
So now, this is really where we get creative, and we start thinking.
00:42:15.700 --> 00:42:19.300
What is going to be able to get us the answer here?
00:42:19.300 --> 00:42:25.300
What is going on here--what is the pattern 2, 5, 10, 17, 26...?
00:42:25.300 --> 00:42:28.700
And this is the part where we lean back, and we just think for a while.
00:42:28.700 --> 00:42:34.900
We think, "What does this look like? What have I seen that looks even vaguely similar to the way this grows...
00:42:34.900 --> 00:42:37.100
the fifth term is 26...how does this work?
00:42:37.100 --> 00:42:40.100
We might try coming up with some pattern the first time that doesn't end up working.
00:42:40.100 --> 00:42:45.800
That is OK; the important thing is just to keep looking and keep going, pondering and thinking: how is this related to something else?
00:42:45.800 --> 00:42:50.700
Remember: we talked about some of the other patterns that are likely to show up--squares, cubes, factorials...
00:42:50.700 --> 00:42:53.800
Those are good ones to start trying out if you can't figure it out yet.
00:42:53.800 --> 00:42:56.000
So, let's look at squares: what are the squares?
00:42:56.000 --> 00:43:02.500
Well, the squares would end up going...if we had simply n², then that would end up giving is the sequence:
00:43:02.500 --> 00:43:14.300
1 (1² is 1), then 2² is 4, then 3² is 9; 4² is 16; 5² is 25...
00:43:14.300 --> 00:43:21.100
How does 1, 4, 9, 16, 25 relate to 2, 5, 10, 17, 26?
00:43:21.100 --> 00:43:25.100
Oh, they are very similar; we are just adding 1.
00:43:25.100 --> 00:43:36.700
If we add 1, +1 would get us 2; +1 would get us 5; +1 would get us 10; +1 would get us 17; +1 would get us 26.
00:43:36.700 --> 00:43:41.000
We have figured out what the general term here is--what the nth term is.
00:43:41.000 --> 00:43:51.900
It is the formula a<font size="-6">n</font> = n² (because it is the number squared), but then we also have to add 1: a<font size="-6">n</font> = n² + 1.
00:43:51.900 --> 00:43:56.800
That seems to give us the formula for our general term for this sequence.
00:43:56.800 --> 00:43:58.900
Let's check and make sure that that is, indeed, the case.
00:43:58.900 --> 00:44:06.800
We do a quick check; if we plug in a₁ = 1² + 1, then we get 1 + 1, which equals 2.
00:44:06.800 --> 00:44:13.900
That checks out; if we plug in for our second term, a₂, then we would have 2² + 1.
00:44:13.900 --> 00:44:18.300
2² is 4; 4 + 1 is 5; that checks out.
00:44:18.300 --> 00:44:25.800
Our next one: a₃ = 3² + 1; 9 + 1 = 10; that checks out; great.
00:44:25.800 --> 00:44:33.300
At this point, it seems that our n² + 1 ends up working out; it ends up following this sort of like the squares method,
00:44:33.300 --> 00:44:38.800
but a little bit more, just adding one each time; we see that the pattern that we figure out ends up being the same.
00:44:38.800 --> 00:44:43.100
Notice: this pattern isn't really so much a pattern about a recursive thing going on.
00:44:43.100 --> 00:44:49.900
It is not really so much about adding the same number each time, multiplying by some number each time...
00:44:49.900 --> 00:44:54.700
We could figure it out as a recursion formula, but it is easier to think of it just in terms of this absolute:
00:44:54.700 --> 00:44:58.600
here is the general term; here is how any given location ends up working.
00:44:58.600 --> 00:45:02.200
It is the number of the location, squared, plus 1.
00:45:02.200 --> 00:45:08.400
All right, the fourth example: Given the recursion relationship below, write the first five terms; then give the nth term.
00:45:08.400 --> 00:45:13.600
We have a<font size="-6">n</font> = -3 times a<font size="-6">n - 1</font>; that is to say, some term
00:45:13.600 --> 00:45:22.400
is equal to -3 times the previous term; a<font size="-6">n</font> is some term; a<font size="-6">n - 1</font> is the term 1 back, going backwards by 1.
00:45:22.400 --> 00:45:27.000
So, it is -3 times the previous term; and we have to have some starting place to begin with;
00:45:27.000 --> 00:45:30.300
otherwise we won't be able to always look at previous terms.
00:45:30.300 --> 00:45:36.100
So, we start at 2; then, the next one...if we want to talk about a₂, the second term,
00:45:36.100 --> 00:45:45.200
it would be -3 times a₁, so a₂ is equal to -3 times...a₁ is just 2, so times 2; that equals -6.
00:45:45.200 --> 00:45:51.200
So, a₂ = -6; notice: all that we really did there was just multiply by -3.
00:45:51.200 --> 00:45:54.000
So, we can probably just end up doing this in our head.
00:45:54.000 --> 00:45:58.400
2, then -6; what would come after that?
00:45:58.400 --> 00:46:04.700
2, then, -6; then we would multiply by -3 again, -3 times the previous term, to get our next term.
00:46:04.700 --> 00:46:16.900
-6 times -3 gets us +18; to get the next term, 18 times -3 is going to end up being -54, because it is -3.
00:46:16.900 --> 00:46:22.200
The next term is going to end up being...times another -3...positive 162.
00:46:22.200 --> 00:46:27.600
And it will continue on in this matter; so we have now figured out the first 5 terms.
00:46:27.600 --> 00:46:32.500
Great; but we also have to give the nth term, so how can we figure out what the nth term is?
00:46:32.500 --> 00:46:41.600
Let's switch colors for this; the nth term...notice: 2, -6, 18, -54, 162...it is kind of hard to see a pattern there really obviously.
00:46:41.600 --> 00:46:48.000
We see that every time, it is multiplying by -3, because we were told very explicitly that the recursion formula says to multiply by -3.
00:46:48.000 --> 00:46:52.000
So maybe we could make that -3 show up so that we could see that a little more easily.
00:46:52.000 --> 00:46:57.700
If we do that, we could write its 2 first; we have 2 show up at the beginning.
00:46:57.700 --> 00:47:00.200
It doesn't have any -3s multiplying by it yet.
00:47:00.200 --> 00:47:07.100
But next is going to be (-3)¹ times...well, we wouldn't be able to figure out immediately that it is going to be to the 1.
00:47:07.100 --> 00:47:16.700
So, -3 times 2; then the next one would be -3 times it again, so now we have (-3)² times 2.
00:47:16.700 --> 00:47:23.200
And the next one would be (-3)³ times 2, and the next one would be (-3)⁴ times 2.
00:47:23.200 --> 00:47:26.300
And it would just continue on in this manner.
00:47:26.300 --> 00:47:29.600
So remember: we want to get the whole thing to look like a similar format.
00:47:29.600 --> 00:47:34.100
Everything has these -3's on it, raised to some exponent, for the most part.
00:47:34.100 --> 00:47:40.700
Most of them have the exponents, and all of them end up having the -3 business here,
00:47:40.700 --> 00:47:44.100
with the exception of this one, which has neither exponents nor -3.
00:47:44.100 --> 00:47:46.700
So, we need to get them to all have this similar format.
00:47:46.700 --> 00:47:52.100
We have -3 to the what?--what number can we multiply by?--what can we always multiply by?
00:47:52.100 --> 00:47:58.900
We can always multiply by -3 to the 0; it is (-3)⁰ times 2, because that is just 1.
00:47:58.900 --> 00:48:12.900
Over here, we have -3 to the 1, times 2; -3 to the 2, times 2; -3 to the 3, times 2; -3 to the 4, times 2; and so on.
00:48:12.900 --> 00:48:23.100
So, if that is the case, we can match this up to n =...1 is here; n = 2 is here; n = 3 is here; n = 4 is here; n = 5 is here...
00:48:23.100 --> 00:48:30.600
What changes each time? The only thing that ends up changing is 0, 1, 2, 3, 4...everything else is the same.
00:48:30.600 --> 00:48:36.600
n = 1 gets us 0; n = 2 gets us 1; so what are we doing to the n? We are subtracting 1 each time.
00:48:36.600 --> 00:48:47.800
So, we see that the general term can be given as a<font size="-6">n</font> = -3 raised to the n - 1 times 2.
00:48:47.800 --> 00:48:54.000
And there we go; if we want to check that, let's just check for the third term.
00:48:54.000 --> 00:49:04.500
Just randomly, to make sure that everything ends up working out: the third term, a₃, is equal to -3, raised to the 3 - 1 times 2.
00:49:04.500 --> 00:49:16.300
That is -3...3 - 1 is squared, times 2; -3 squared equals 9, times 2...9 times 2 is 18.
00:49:16.300 --> 00:49:19.400
And that checks out with what we already figured out is the third term.
00:49:19.400 --> 00:49:24.700
Great; so we figured out our general term, our nth term formula; it works out perfectly.
00:49:24.700 --> 00:49:31.000
The fifth example: Find the nth term for the sequence below; assume that the sequence starts at n = 1.
00:49:31.000 --> 00:49:39.300
The first thing: most of it ends up being in this fraction, fraction, fraction, fraction...not a fraction.
00:49:39.300 --> 00:49:44.400
We want to get everything in terms of these fractions, so let's get everything into fraction format.
00:49:44.400 --> 00:49:51.500
1 + 2 over 1 is how we will replace that; and then, 2 + 3 over 2, and so on and so on.
00:49:51.500 --> 00:49:54.000
The rest of them are now in fractions.
00:49:54.000 --> 00:50:03.500
We can write this as...here is our n = 1; here is our n = 2; n = 3; n = 4; n = 5; n = 6.
00:50:03.500 --> 00:50:12.200
Notice: everything ends up changing; all of the numbers in each of these terms end up being different from the previous and the next term.
00:50:12.200 --> 00:50:21.900
But we end up seeing some connections here: 1 matches to the 1's here; 2 matches to the 2's here; 3 matches to the 3's here;
00:50:21.900 --> 00:50:28.500
4 matches to the 4's here; 5 matches to the 5's here; 6 matches to the 6's here.
00:50:28.500 --> 00:50:32.300
And we see that this other number is just + 1 each time.
00:50:32.300 --> 00:50:40.200
1 + 1 gets us 2; 2 + 1 gets us 3; 3 + 1 is 4; 4 + 1 is 5; 5 + 1 is 6; 6 + 1 is 7.
00:50:40.200 --> 00:50:43.500
So now, we have an easy way to figure out what the nth term is.
00:50:43.500 --> 00:50:49.600
The nth term is going to be equal to a<font size="-6">n</font> =...well, it seems to be...
00:50:49.600 --> 00:51:00.600
this one is just our value of n; plus some fraction...the bottom is also the value of n, and the top is n + 1.
00:51:00.600 --> 00:51:05.600
And there we go: do a quick check, because it is always a good idea to do a check.
00:51:05.600 --> 00:51:18.100
Let's check it out: a₁ would be equal to 1 + 2, over 1; so we get 1 + 2...that checks out with what we already had as the first term.
00:51:18.100 --> 00:51:27.400
If we wanted to try another one, like a₂ = 1 + 2 + 1, over 1, which would be 1 + 3, over 1...
00:51:27.400 --> 00:51:35.900
oops, sorry: not over 1; not over 1; it is divided by n as well; sorry about that mistake; divided by 2; 1 + 3 over 2.
00:51:35.900 --> 00:51:40.100
Oh, I did it on both of them; it is important to end up using your formula in the check.
00:51:40.100 --> 00:51:48.300
It is also an n here; 2 + 2 + 1 over 2; 2 + 3 over 2; and that does check out with what we had here.
00:51:48.300 --> 00:51:55.300
Great; the last example: Find the nth term for the sequence below; assume the sequence starts at n = 1.
00:51:55.300 --> 00:52:02.000
We see, right away, that this thing kind of changes its format a fair bit in these two.
00:52:02.000 --> 00:52:03.900
It is totally different in these two.
00:52:03.900 --> 00:52:07.600
Before we even really start looking for patterns, we want to get everything into the same format.
00:52:07.600 --> 00:52:09.900
That will just make it easier to see patterns.
00:52:09.900 --> 00:52:12.100
So, how can we get them into the same format?
00:52:12.100 --> 00:52:14.900
We first certainly need to have them as fractions.
00:52:14.900 --> 00:52:30.900
So, as fractions, we have 1; we can always divide by 1, so 1/1; then 3/1, and then 3²/2, 3³/6, 3⁴/24, 3⁵/120.
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The first thing that you are probably noticing is that we have 3⁵, 3⁴, 3³, 3², 3...
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Well, we could write this as 3¹; how can we write 1 out?
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Well, remember: 3 to the 0 equals 1; any number raised to the 0 comes out to be 1.
00:52:46.400 --> 00:53:11.800
So, we could rewrite the top as 3⁰/1; 3¹/1; 3²/2; 3³/6; 3⁴/24; 3⁵/120, and so on.
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At the top part, we now see pretty clearly that there is a pattern.
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It is that the number increases by 1 each time from the exponent on the 3; it starts at 0, so that would be 3^n - 1.
00:53:20.900 --> 00:53:27.700
But what about this bottom part, 120, 24, 6, 2, and then we have 1 and 1 here...?
00:53:27.700 --> 00:53:35.600
1, 1, 2, 6, 24, 120...well, if we worked backwards, we might recognize factorials.
00:53:35.600 --> 00:53:40.100
Remember: we talked about factorials earlier in the lesson; that looks like factorials.
00:53:40.100 --> 00:53:47.700
So, 5 times 4 times 3 times 2 times 1 is 120; 4 times 3 times 2 times 1 is 24; so we have 5! on the far right...
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We can write this as 3⁵/5!; we will keep going, so let's work backwards...3⁴/4!;
00:53:59.700 --> 00:54:13.700
3³/3!; 3²/2!; 3¹/1!...that would just get us 1.
00:54:13.700 --> 00:54:21.600
And here is the most confusing part of all: 3⁰ over...well, what can we do that will end up having factorials involved, and still get us 1?
00:54:21.600 --> 00:54:25.400
Now we have to go back and remember: there is a very specific thing about factorials.
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The way that factorials work is that 0! is just defined to equal 1.
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So, that means we could also write this as 0!; so we have maintained a pattern.
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We are going to end up seeing patterns, because all of these problems are going to be based on patterns.
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So, we know that it is a factorial pattern; it is no surprise that it is going to keep going.
00:54:42.500 --> 00:54:51.100
We have on the bottom 0!, 1!, 2!, 3!, 4!, 5!...on the top: 3⁰, 3¹, 3², 3³, 3⁴, 3⁵.
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Let's compare that to the numbers n = 1, n = 2, n = 3, n = 4, n = 5, n = 6, our first location, our second location, third location, etc.
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With that in mind, we see that the top exponent is always equal to the value of the location.
00:55:10.900 --> 00:55:15.100
At n = 5, we get a 4 exponent, so it is always minus 1.
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Similarly, the 0!, 1!, 2!, 3!, 4!, 5!...it is always the number of the location, minus 1, to get to the number in the factorial.
00:55:27.200 --> 00:55:35.600
In the third location, at n = 3, it is a 2! (3 minus 1); in the sixth location, it is a 5!, 6 minus 1.
00:55:35.600 --> 00:55:43.600
So, we see that they are both based off of this n - 1 business; so that means we can finally set our a<font size="-6">n</font> equal to...
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it is 3^n - 1, over (n - 1)!; and there is our answer.
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There is the general term; and it is always, always a good idea to check your work with this sort of thing;
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So, let's just do a quick check to make sure that this ends up coming out.
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At a₁, we would have 3^1 - 1/(1 - 1)!; so that is 3⁰/0!, so we get 1/1,
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which equals 1, which checks out with what we initially had.
00:56:21.900 --> 00:56:26.400
Let's try jumping forward to a slightly larger number, so we can check against something else.
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At a₄, we would be at 3^4 - 1/(4 - 1)!; so we have 3³/3!.
00:56:40.300 --> 00:56:45.800
Everything is in this 3³, so we don't have to simplify that to 27; we can just leave it as it is.
00:56:45.800 --> 00:56:54.100
3³/3!...that comes out to be 3 times 2 times 1, or 6; so 3³/6 is what we have for our fourth location.
00:56:54.100 --> 00:57:00.200
1, 2, 3, 4th location: 3³/6; that ends up checking out.
00:57:00.200 --> 00:57:03.000
We end up seeing that our general term makes sense.
00:57:03.000 --> 00:57:07.500
All right, we have a really good understanding of how sequences work, and how we can get general term,
00:57:07.500 --> 00:57:10.300
nth term, formulas from looking at them for a while.
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Recognizing patterns is a really useful skill; it will show up in a whole bunch of different things in math.
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Even if you end up thinking that this is a little bit difficult now, trust me: it is going to end up paying dividends later on.
00:57:20.400 --> 00:57:23.600
You are going to end up using this stuff a lot, finding patterns in a variety of stuff,
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whether it is in science class, math class, economics...whatever you end up studying.
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You are going to end up having to find patterns of some sort.
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Even in English, patterns are really important things; you are going to talk about themes in a book, so patterns really matter.
00:57:36.500 --> 00:57:38.000
This sort of thing is really important.
00:57:38.000 --> 00:57:40.900
We have a good understanding for sequences; now we are ready for the rest of this section.
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We will have a whole bunch of ideas, working from this base of sequences.
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All right, we will see you at Educator.com later--goodbye!