 Hey, hello friends, welcome to this session on sequence and series and we will be discussing one variety of sequences in this particular session and a few subsequent session and that particular variety is called arithmetic progression. So we will understand this in little detail and we will understand this type of sequence in more details and understand its behavior, its properties and you know, learn different aspects of the same. And in the previous session, if you recall, we were talking about functions or you know, the sequence whose nth term can be expressed in a form of a function of n itself, right? So in the previous sessions, if you remember Tn was given by fn, where fn could be linear, quadratic, cubic and you know, multiple such other varieties, for example, if you remember Tn, for the first case, we discussed like this a n plus b. In the second case, in the second case, the function was a quadratic function on n, a n square plus b n plus c. Then we had you know, Tn as a cubic function a n cubed, a n cubed, a n cubed plus b n squared plus cn plus d, like that, isn't it? And depending upon the sequence, we used to find out the values of a, b and c, which are rational numbers. So that is what we studied in the previous few sessions. Now, henceforth, we are going to analyze one particular variety and this is this linear one. And this is what we study under the heading arithmetic progression. Okay, so now you know what is arithmetic progression. Any sequence, any sequence, which whose nth term, whose nth term can be expressed as a linear form function of n will be called, will be called a linear arithmetic progression. So as the name suggests, there is a progression. Progression is what progression is continuous, you know, change. Yeah, it can be, you know, positive as well as negative progression. That means it can increase as well as decrease. Okay, so that is what is progression. And, you know, since it is, you know, a linear function, it's, you know, the increase, the rate of increase is always same. So it increases by the same amount every time. What do I mean by all of this? So let's say we have sequence like three, six, nine, all multiples of three, let's say, okay, like that. And if you see this particular sequence is, you know, increasing at a constant rate. So every time with every, you know, every next seek term of the sequence increases just by three. Okay, so, right, so we attribute it as arithmetic progression. Okay, so what all our different examples of arithmetic progression, so all natural numbers, so all natural numbers one, two, three, four, like that to any N is arithmetic progression. Similarly, like, all even let's say even numbers are arithmetic progression are in arithmetic progression. All odd numbers are in arithmetic progression. Why are they in arithmetic progression? Because if you look closely, the difference between any two consecutive numbers, so hence we are saying any two consecutive terms, so Tn plus or Tn minus Tn minus one, okay, Tn minus Tn minus one that is nth term. And it's, you know, a term just behind it. If you find this difference, this is coming out to be constant. This will always be constant. Okay, similarly, you can also say Tn plus one minus Tn, that is term, nth term and the next term. If you take the difference, it will come out to be constant. That is the beauty of this particular type of sequence. And these sequences will be called arithmetic progression, right? So again, generalizing, I can say T2 minus T1, the difference between any two term, any two consecutive term, and T3 minus T2, this is again, consecutive terms, T4 minus T3, this is equal to T5 minus T4 and this will go on, go on, right? This is T6 minus T5 and so on and so forth. You can check any, you know, for all these, for all these sequences, this particular thing happened, you know, is valid. That is the difference of the subsequent terms is constant. For example, in this case, it is one. In this case, it is two. In this case, also it is two. And in this case, it was three. Right? Now, this particular difference is called common difference guys, what is it called? Common difference, common difference. And we have given them, or this particular term, a name, that is D, or a notation D. So Tn, n plus one minus Tn is D. Okay. And since I said that Tn is given by a linear function a n plus b, let's say where a and b are constant, a and b are constant. Okay, so what will be in the next term in this sequence, in the sequence Tn plus one, if you replace n by n plus one, what will you get a n plus one plus b simply replace n by n plus one, you will get a n plus a plus b. Let's say this is equation number two, this is number one. And if you do two minus one, two minus one, you will get what Pn plus one, sorry, n plus one minus Tn, isn't it? This is D, common difference. And this will be equal to a n plus a plus b Tn plus one was this, and minus Tn will be a n minus b. So if you see this, and this, and this goes, so you get a, which is a constant, right? So it independent on the value of n, you get a, right? Such sequences will be called or arithmetic progression, always keep in mind. So what are arithmetic progression guys? So any sequence of numbers where the conjugative, you are the difference between two conjugative terms is constant will be called an AP. Okay, I hope you understood this part. Now, having understood the concept of AP or the definition of AP, now let's take few examples and problems. So for example, the question is show, show that, show that the sequence, sequence, so that the sequence defined by, defined by Tn is equal to two n square plus one is not an AP. Okay, show that the sequence defined by Tn two n square plus one is not an AP. So how do we do it? Let's first find out the AP itself. So what is T1? Let's solve this problem. And we'll show that this is not a sequence, which is an AP. So T1 will be simply two times one square plus one, which is how much two plus one three, correct? T2 is equal to two into two square plus one. So how much it will be two into two square is eight plus one nine. I hope this is understood, right? Eight plus one nine. What is T3? So two into three square plus one, which is nine to 18 plus one 19, right? Then T4 is two into four square plus one, that is 32 plus one 33. Now here is the thing. So if you find out T2 minus T1 for AP, what do we need? Just to prove that the conjugative terms are having constant difference. But let's try and find out, find it out. So T2 minus T1 is nine minus three, which is six. And T3 minus T2 is 19 minus nine, which is 10. Okay. And we get T4 minus T2 is 33 minus 19, which is 14. Now we are seeing that they are not constant, right? So these are not, these are not constant. So since, since T2 minus T1 is not equal to T3 minus T2 is not equal to T4 minus T3. Hence we can say this sequence, this sequence is not an AP, is not an AP. Okay. Because for AP, the first criteria is this must be equal consequence, you know, difference between conjugative terms must be equal. So this, since this is not there, so hence it cannot be represented as a, or it is not an AP. And we had learned in the previous session also, this is a quadratic form guys. So for, for AP, we need a linear form. Is it it? So hence it's not an AP. Here is another question guys. The question says the nth term of a sequence is three and minus two. First part is, is the sequence an AP? You have to find out if the sequence is an AP. And if so find it's 10th term. Okay. So let's solve this. And by appearance, it looks like it's an AP because nth term is given by a linear term, linear function three n minus two is a linear function in n. Okay. So let's try to find out T1, T2 and all that. So T1 is three into one minus two. That is one T2 is three into two minus two, that is four. So T3 is equal to three into we are expecting that the next term should be seven for making it an AP, isn't it? The common difference is three. So three into three minus two is indeed seven. So hence this is appearance wise it is AP. Let's check for one more term. 3.4, three into four minus two is 10. And very clearly T4 minus T3. How much is it three, which is also equal to T3 minus T2, which is also equal to T2 minus T1, all are three. Correct. All are three. So hence we can say that this particular sequence isn't AP. So what are the terms? Terms are one, four, seven, 10, dot, dot, dot, like that. Okay. So hence it is, it is an AP. AP as, as the consecutive terms, the difference, the difference of two consecutive, consecutive, let me write it here, consecutive terms, the difference of two consecutive terms is always constant. Okay. Okay. Right. It's always constant. This is why it is an AP and 10th term, hence 10th term, we have to find out 10th term. 10th term is nothing but T10, which is nothing but what, what was of a relation given? It was given as three N minus two. Is it a three N minus two? Yeah. So hence it is three into 10 minus two. So 30 minus two, 28. So 28 should be the 10th term of this particular AP. I hope you understood this problem.