 Hey, hello friends welcome again to another session on sequence in series in the previous session We talked about the definitions of sequence definition of series Types of sequences we saw we saw finite infinite sequence and we started understanding the terminologies associated with this particular topic and One of them was to define the nth term. So we used to define nth term like that. So one notation was of The nth term right and we express this as Tn and this is nth term that is a term which is positioned at nth place or Right, so example, let's try and understand using an example. So we said If the sequence is something like 1 3 5 7 and 9 so odd integers are there then this is certainly t1. So this is t1 This is t2 Right, this is t3, right? So this subscript this susk subscript represents an index or the position correct or the position of the particular term from the beginning I Hope this is clear and this term and terminology. We are going to use multiple number of times the same terminology is similar to an or Un as I discussed last time also so anything can be used to denote a nth position Number in that particular sequence, right? Now once that is clear Then we are going to discuss about, you know, two types of sequences in this case So let's under understand other typeification. So last time we did finite in finite series Here are sequence here. Let's say one is when the nth term is a direct function of n that is you can have a formula kind of a thing formula actually it's a relation so formula slash let's say relation in n in n which which determines and determines Pn nth term, right? So Pn is a function of fn. So example, so let's take Example, Tn is given by let's say n itself the most basic example could be Tn is equal to n itself So what are the elements or are these you know members of this sequence so you can see when n is equal to 1 Tn will be 1 Right because Tn is equal to n itself if n is equal to 2 then Tn will be equal to or T so let's write the Subscripts as well so T1 T2 so T2 is 2 then for n is equal to 3 P3 is equal to 3 itself and if you go on doing this you'll get for any n Pn is simply n and what are these? This is nothing but all natural numbers if you see all natural numbers isn't it so hence by this particular expression or you know this particular relationship I can Represent the entire natural number sequence another examples. Let's take another example. Let's say Pn is equal to 2n plus 1 okay, so Let's start again. So n is equal to 1 if n is equal to 1 then T1 will be how much 2 times 1 plus 1 Correct. What are we doing? We are simply replacing this n by the given number in case case 1 Right, so hence. What is it? It is 3 What will be n is equal to 2 so if n equals to 2 so T2 is 2 into 2 Plus 1 and hence it is 5 Correct for n is equal to 3 T3 will be 2 into 3 plus 1 that is 7 and so on and so forth you can you know So let's say now someone asks you a find the 10th Number in the sequence right 10th number n is equal to 10 So 10th number that means what will be the 10th number in the sequence So I don't really need to go one by one to each one of the you know Terms in the sequence to find out T10. I can directly deploy the formula 2 into 10 plus 1 is 21 Right, so the 10th term will be 21. Similarly if someone asks find out 100th number So T100 I don't need to calculate one by one each at a time I simply deploy this and one so it is do not one right, so this is the beauty of you know this Sequence that if you have a Relationship known you can find out Any position number Okay, any position number. This is one way of expressing Or let's say one one type of sequence another type So this was type a if you see this was type Let's say this is type a where Pn is a function of the n itself, right now What happens in the other type type B is nothing but the nth term is a function of previous terms Okay, so Tn is a function of let's say Pn minus one Pn minus two or however possible, right? Very simple. How can you you know? Let's take an example then so let's say Tn is equal to two times Tn minus one so I hope you understand what is Tn and Tn minus one so in any given sequence for example one three five seven nine This if this is Tn Then this becomes Tn minus one that is what it means Okay, if this is Tn Then this becomes Tn minus one one term prior to Tn Tn, right? So, let's say if we can start with this we start with certain assumption. Let's say Tn T1 is equal to one Let's say say assume we are now generating a new sequence So hence what will be T2 if you see T2 is two times T1 From this relationship and T1 we already know so two into one we had assumed it to be one So hence it is two so second term of this sequence is two Let's calculate the third term of the sequence third term of the sequence would be twice T2 simply right now. So hence two Into T2. What is T2? Two. So I'll get four Let's calculate T4. So T4 is two into T3 one previous term. So hence it is two into Four this is eight so on and so forth. So hence any Tn for that matter will be simply two into Tn minus one Correct. This is this is one way of You know expressing the terms of the sequence correct. So hence in this sequence you are getting one two Four eight then next will be sixteen next will be thirty two and so on and so forth Okay, another very famous one in this type is the famous Fibonacci sequence Fibonacci Sequence yep Fibonacci sequence. It's around One million in AD around that time Thousand 80 or thousand 1100 AD around that time this guy this Italian person Mathematician Fibonacci he came up with this kind of a series one sequence another one one two and three and then five and then eight thirteen 21 34 55 89 and so on and so forth. So if you notice this sequence carefully, you will see that any term Tn and When n is let's say greater than two so for any term greater than two, right? Right any n greater than two Tn is simply Tn minus one plus Tn minus two right see the NF term is some of the previous two So let's take random any term here. So if it is if you see 21 21 is made up of eight plus 13 check Correct if you take eight eight is some of three plus five Correct 55 is some of 21 plus 34 Okay, so this is another type of sequence so we can you know We can get two types of or let's say all the sequences can be categorized into two Types one is the direct function and where the nth term is function of n and another one is Where where the where the nth term is function of The previous terms and it could be any combination of previous terms not necessarily two or one or three Depending upon we again typify all these type of Functions these are also called recurrent relations and you will understand about this in higher Grades and then you will be analyzing more. You know these of these more in detail later on Okay, it's not that if some function is a you know or let's say if nth term is a function of or nth term is a you know dependent on previous terms we can't have this kind of a relationship for this meaning what the same Same term which is a function of Previous terms can also be function of in terms of n so there can be a mechanism by which we can convert This type of relationship into direct function of n that is another you know Topic of discussion so we will be taking it up as we progress in this session So thanks for watching this session and I hope you understood these two types in the subsequent session What we are going to do is we are going to focus on first category of this type of Sequences more and we'll see if a given sequence is given can we come back and find out This function so that we can predict any nth term in that sequence that we'll see in the next session