P.A - PROGRESSÃO ARITMÉTICA AULA 01 \Prof. Gis/ Termos e Razão de uma PA

In this class you will learn arithmetic progression, our famous PA, but what is a PA? An arithmetic progression? Oh, remember, every time I say PA you will remember arithmetic progression.

So let's see the definition that I brought to you today: it is a sequence of numbers in which each term from the second onwards is equal to the previous one added to a constant called ratio. I didn't understand anything, Gis do Céu, what that definition is, I can't understand, calm people. So let's go to Gis's way of explaining it to you, let's do it very calmly and explain this definition of what an arithmetic progression is.

So look at this numerical sequence that I brought you, can you observe any pattern in this numerical sequence? Take a look, it’s easy, right? It's pretty easy.

If you look at how many units were added from 2 to 5? it added three, right, so here plus three from five to eight it added three from eight to 11 it added three so if I wanted to find out the next term I should also add three which would be who the next one would be 14 and so on. So see that here I have an arithmetic progression.

Does it make a little more sense now that this definition is each term starting from the second but because from the second, because the first I have already done here there is one from this here I can already observe the pattern that was added o 3 o, is equal to the previous one, so if I take this second one here o is equal to this previous one added to a constant and this constant is called ratio, but reason is when you are right, people, the reason I refer to here Oh, I don't have this pattern here, this number that I added in this case, this arithmetic progression that I have and other observations that we are going to make here in this sequence so we have the first term, didn't you watch Gis's class on numerical sequence? So take the indication and go back, because I explained first term second term, how is the representation? So this number two this number two would be our first term and to represent it the first term of a sequence I call it A1 so A1 is 2 which is the first term, that's why it's important to see the previous class folks, then what I have here in this PA I have the reason I'm going to call the reason here R it's abbreviated to R the reason is 3 right and here in this case I don't have the number of terms because this PA is what even when There are these.

. . in an infinite number of sizes.

So in this case, just by looking, I don't have, I don't know how many terms this PA has or who the last term is, so here I can get these two pieces of information by looking at it because here it's not an exercise to solve just to take a look. So see here in this case we have an infinite PA because one thing that you can see is infinite right away is this. .

. here, ok guys? But will there ever be a case where the PA will be infinite?

Come on guys, shall we go? So now take a look at these other three examples that I brought here, so there are three numerical sequences, I'm saying it like this because you need to pay attention. Sometimes there is a trick when you are going to do an activity, a test, sometimes there is a numerical sequence and this numerical sequence is not an arithmetic progression.

But how will I know that? Well, you will have to identify what that constant is, the ratio, so from the second term you will have to analyze whether this second term here was obtained from the sum of the first one there with the ratio, so keep an eye on the ratio and always that pattern were we really have a PA if it is not just a numerical sequence, does this difference have to be marked then agreed? So in this case here, do we have a PA or just a sequence?

shall we analyze? There are some that are very easy, right guys, I look here, from 1 to 3, from 1 to 3, two were added together, right? From here, from three to five, from three to five added two, from five to 7 added 2, from 7 to 9 added 2, and another way for you to find out the reason if you look there for example and you can't, oh, it's easy, but you can't identify which constant is being added.

You can make a difference, take the term in front and subtract 3 - 1= 2, 5 - 3= 2, 7 - 5= 2, 9 - 7= 2 does inverse operation So it's clear that this is what I'm going to use here, as you can see there is a negative result in the sequence, so let's keep an eye on it. So what can I mark from here so I have an arithmetic progression yes and the first term our A1 in this arithmetic progression is a ratio, the ratio is 2 and what's more I can now extract information about how many terms we do we have in this PA? We have one two three four five terms so the number of terms will be represented by the letter N number of terms so N = 5 and as you may have seen in my numerical sequences class I talked about a term called AN which would be the term of my sequence like here I have a finite PA so I have its last term, so my AN, actually there is not AN, because I already know who this AN is, which is the last term, so just to remind you AN It's going to be 9 and remembering that this N here is this five, I'm just going to leave it identified as AN so you can memorize it, because when we use the formula you'll always be talking about this AN here, so take advantage and mark it, okay?

And then this AN is finite or infinite? iIt is finite, right, we don't even have the. .

. there and it ended at 9, so here we have a finite, finite PA, ok. Let's analyze this other case here, what is happening here is it a numerical sequence and is it a PA?

Let's analyze from 6 to 6 added 0, from 6 to 6 added 0 and from 6 to 6 added 0 and as I said if you want to know the reason take the one in front and subtract 6 - 6= 0 reason is zero right, Is it a PA? Is it a PA, did we follow a pattern here? Do I have a finite or infinite PA?

Can you identify it yet? I have a finite PA because it has an end, it stopped here, right at 66, so let's identify like we did here, A1 who will be A1? o 6.

Who will be the reason in this PA that I brought? the ratio is zero who will be on what is on the same on is the number of terms how many terms do we have one two three four, the number of terms is 4 and the AN which would be A4, right which is my last term here it will be six. Okay, if you want to change it here, instead of writing the AN for A4, that would be A4 because I A1 A2 A3 A4 already change it, but memorize this AN here well, so in the exercises you will be saying some AN, okay?

And this PA, we already said, is a finite PA just like the one above, let's write it here, finite. And in this other case, who can identify whether this is just a numerical sequence or is it classified as an arithmetic progression? Can you identify the reason?

How much was added from this term here to this one? Can you already identify when it is negative? Hmm, you're having difficulty, right?

You got the negative, people are already having difficulty. So do what I said, take the one in front and subtract it, calculate the difference between the terms, do it like this, then take the - 11 and subtract it, take this - 11 and subtract it from the one at the back, then subtract the one at the back which is minus 7 Oh, and here's where people get confused, subtract from minus 7, it's minus and minus, then you're going to do the sign rule here, so here, minus with minus is plus, it's going to be minus 11, let me write it again - 11 plus 7 if I owe 11 and I have seven I owe four so it means here that the ratio is - 4 so minus four. So that means I added -4 and here I added -4 the reason is -4.

It's OK? let's test negative three with - 4= -7, - 7 with - 4= - 11 then you could have identified it without doing this calculation here. But if you are in doubt, does this difference take the front term and subtract it from the back term, and then I have the one in my arithmetic progression because we saw that it is an arithmetic progression, it is negative three, the reason we saw that it is - 4 what more than we have here we have the amount of terms we don't have, right because it's an infinite PA.

So I don't know, I 'm just going to calculate the number of terms or who the last term is when I have a statement or an exercise. So for now let's take it easy and I have here an infinite PA, ok infinite, infinite. So, another thing I want you to notice, besides everything I've already said is that it's a PA, it can be classified as increasing, decreasing and constant, how am I going to identify a PA, how am I going to classify a PA, so, you will look at the ratio so if your ratio is positive, as is the case here, we will have an increasing AP, so let's write here, PA, PA is increasing and why are you going to justify that this PA is increasing?

because the ratio is positive then I will write R is positive? How do I identify this? How do I write mathematically that the positive number is because it is greater than zero, so R is greater than zero, remember this well, the reason I go with one + so you don't forget that, here the reason is positive, it is greater than zero, you will have a growing shovel.

In this case here we have a ratio that is equal to zero so you will have a constant BP. So let's just write in this case PA, it will be called constant, so constant is when its ratio is equal to zero, ok? And then in our last case we won't have what in this last case we will have a PA which is the one that is missing, right because the students go by elimination, decreasing lack, right and you can see here that the reason for our PA is negative, so I have a decreasing BP and the justification that it is a decreasing BP is because the reason is less than zero, less than zero because it is?

make a draft, it's negative. And you can also just classify the PA by looking at the sequences, you don't need to look at the reason, if you can identify here the values ​​are increasing, here it remained constant, it didn't change and here what's happening is what's happening, the values ​​are decreasing, we pay close attention when we have the negative ones, yes, it is decreasing because the negative three here is greater than the - 11 because the negative three is closer to zero on the number line, didn't you attend Gis's class on the number line with negative numbers ? so as not to get this content wrong here, ok?

Did you understand correctly? So, print the screen of our PA classification and be sure to watch the next classes as I will teach you how to do the exercises and solve exercises based on the general term formula, people, so don't miss the next class and also take advantage now to subscribe to Gis's channel, leave a super like if you liked the class and share this entire class with your colleagues and see you next time!