 Welcome friends. We are going to discuss a new topic in this session and the topic's name is sequence and series. In the subsequent sessions we will be covering the entire important aspects of sequence and series up to let's say senior secondary level and the first few sessions will be mostly towards introduction and making you familiar with the terms and terminologies used in the particular topic and then we will be analyzing the different behavior and properties of sequences in series. So like any other topic we have to first understand what sequence and series are and before that also we just need to figure out what are the application areas why at all we are even studying these topics right where in which fields in mathematics and let's say in our day-to-day life we get to you know see sequences and series okay. So before we start let's talk about what we have already covered or what we have understood or you know what we have gathered knowledge about so far in you know in practical world. So one such area is the area of interest right so interest in the sense the commercial interest what we are talking about simple interest and compound interest and things like that. So how simple interest was calculated and how these are linked to let's say sequence and series and for that matter how compound interest was calculated and how there you know there is a linkage between that particular topic and this one. This is being done to give you a overview of how you know the practical aspects of the topic which we are going to study so that there is a connect between the real-life situation and whatever is the you know field of study here. So let's talk about simple interest first so if you remember your previous grade concepts you know how to calculate principle or simple interest right. So simple interest works like that that if you lend a amount P principle amount P for let's say one year one year at a rate of at interest rate we say interest rate of let's say our percentage our percentage so this is how the principle or the total amount is calculated after the given year right. So after one year the person who has borrowed the money will have to pay this much back P 1 plus R E by 100 isn't it after one year correct this much will be the amount he has to pay after one year right. If this is let's say after one year one year at the end of one year this is the amount he has to pay let's say he doesn't pay at the end of first year and he pays at the end of two year then the amount he will pay would be P times 1 plus 2 R T by 100 I hope this is clear to all of you then in third year let's say after third year he will pay how much amount is P 1 plus 3 R T by 100 okay now if you see this there's a pattern which is you know emerging out of here so hence after let's say n years n years nth year after the end of n years let's say if I generalize he will have to pay P into 1 plus n R T by 100 isn't it this is the trend now if you see what particular behavior we observe here so if you take these two amounts clearly the second one second line amount is more than the first one by this amount R T by 100 R T by 100 and again in this case also there is a difference of RT by 100 okay again this also RT by 100 so you can clearly see that you know between any two consecutive years the amount of interest stays the same okay this is our typical behavior or this is what we study in you know this is how is this linked to sequence and series you will come to know that this behavior where you know all these numbers are you know let's say these are a or let me write a 1 a 2 to differentiate a 3 a 4 so if you see a 1 a 2 a 3 a 4 have a relationship between them a 1 a 2 a 3 and a 4 has a relationship and what is the relationship a 2 is a 1 plus RT by 100 isn't it right and then a 3 is a 2 plus RT by 100 like that and if a 4 is nothing but a 3 plus RT by 100 and like that you can find out nth year amount is nothing but a n minus 1 plus RT by 100 this is how the behavior looks like this is a typical behavior of something called arithmetic progression so we will be studying this under sequence and series this is a typical behavior of arithmetic progression where there's a list of numbers and the list of numbers display a particular behavior where the difference between any 2 is constant example of such behavior would be let's say even numbers 2 4 6 8 10 like that so any 2 numbers has a gap of our difference of 2 you can see that correct there could be infinite such examples for example 5 let's say and then 9 then 13 then 17 then 21 so any 2 if you pick up the gap is 4 correct so this is one one type of you know list of numbers so hence you have to remember what we are talking about list of numbers list of numbers and in in in an English literal word literal meaning could be we can write a sequence of number isn't it so these are all these belong to a sequence of numbers sequence of number sequence is what you know list of few items and there is an order in that so when we will formally define sequence you will get to see all this again so the time being understand that there is a list of number and there is some kind of relationship between those numbers okay this is one type let's now talk about compound interest how do they behave compound interest compound interest and you know that in case of compound interest the amount is calculated like a is equal to principal times 1 plus r by 100 to the power n correct where n is what n is the term number term so it could be let's say we are talking about yearly here term is equal to here let's say here okay so how does the you know let's say a1 will be after one year the interest to be paid is or the prince the total amount to be repaid is p by r100 simple this one n is equal to one here then a2 will be simply p1 plus r by 100 to power 2 isn't it after two years he has to pay back this much amount after three years again he has to pay 1 plus r by 100 to the power 3 after four years what will it be after four years the amount to be paid is p1 plus r by 100 to the power 4 isn't it now here also there is you know there is a list of a1 a2 a3 a4 and all that and if you just generalize it you will get an also that is p is equal to p plus 1 plus r by 100 whole to the power n this is the behavior pattern okay now let's analyze this so what is the behavior pattern pattern here if you see a2 upon a1 friends a2 upon a1 if you take this ratio you will get one clear number 1 plus r by 100 then if you take a3 upon a2 then again the same if you know the ratio comes this is surprising isn't it and then a4 by a3 and then a5 by a4 all are same same same to 1 plus r by 100 this is another behavior there the difference was same here the ratio is same the previous case if you see here the difference between the two terms was same here the ratio between the two terms is same and this is nothing but a practical example of a sequence of number which is called geometric progression geometric progression and we are going to study about this also in this chapter okay so what did we learn so here is what we have learned so far we are now analyzing some list of numbers list of numbers are there okay and we are calling them we are calling this list of numbers as a sequence sequence right sequence and there is a you know arrangement of numbers are there arrangement of numbers is there they are arranged in a particular numbers in an order numbers are in numbers in in an order right so order is also you know coming out very clearly here see a1 a2 a3 4 are in order we are not talking about a3 and a7 and a2 and a1 and random numbers like that in order right and if you see there is a there is some kind of well defined well defined law which is common to all well defined law for example here the ratios is same here the difference was same so there's a well defined law as well isn't it so hence these are the three things which we are so here is what we are defining sequences sequence is nothing but a list of numbers we will call that as sequence it must be arranged in an order and there should be some well defined law that will be sequence so and there will be infinite examples of such sequences so all natural numbers form a sequence right example all odd numbers one three five seven right all even numbers two four six eight right all squared numbers one or perfect square one four nine these are all sequences then 16 correct all multiples of five five 10 15 20 all are these examples of sequence all these are example of sequences I hope the sequence is clear the definition of sequence is clear now once the definition of sequence is clear what does series mean then series okay so series is nothing but if you add all the terms of the sequence for example if I find out one plus two plus three plus four plus like that like that you will get a series okay so let's say till n number so whatever is the sum this is an example of series similarly this is example one let's say example two s is let's say sum of all odd numbers okay till let's say 100 okay so all till 100 is a series okay as a series now in in you know so I hope now the clear the two term sequence and series is clear so sequence is nothing but list of numbers again arranged in an order with some well defined law examples are here series is nothing but you add all of them and all the terms of the sequence and then you get the series then we see some types of sequences first of all types of sequences and when there is a sequence you can always expect a series to be associated with it types of sequences so one type is finite series or finite sequence first finite sequence what is finite sequence fixed number of terms if in the sequence there are fixed number of terms it ends actually so we say it is finite and then there is infinite infinite sequence infinite sequence what is it there is no fixed fixed number of fixed number of terms terms terms is not there is not there it is infinite series infinite series right example of infinite series will be let's say one by 2 1 by 4 1 by 8 1 by 16 1 by 32 and so on and so forth goes till infinity okay this is an infinite series finite sequence number of odd or let's say all odd integers less than 1000 all even numbers less than 1000 all perfect square less than 1000 and things like that could be the example of finite sequence so I hope types of sequences finite sequence and infinite sequence is clear to you lots of mathematicians have done a lot of work in these areas and one very famous name was Mr. Ramanujam so very famous mathematician he did a lot of work in infinite sequence and series Ramanujam correct he was a very young mathematician you know could not could not survive much otherwise he would have contributed much to the world of sequence and series okay or mathematics in general now there are few notations you need to understand notations and these notations will be very very useful so let's say any all these and first of all you need to know what is the term of a sequence what is term so any element for example 1 3 5 7 9 11 all of them individually are term so one is a term three is a term five is a term seven is a term nine is a term and 11 is a term and so on and so forth okay we can also say one is the first term is the first term okay first term second then therefore three is the second term three is the second term isn't it then five is the third term five is the third term from the beginning from the beginning that means 11 if you see if you see 11 11 is the sixth term guys sixth term guys right so hence we can have a notation we can say that one is t and subscript one just to notify first element of the first term of the sequence then t2 this could be t3 right and this could be t6 right now so this note this mean this means a term term and this one or two or three or four this this is nothing but an index it gives you the information of the position from the beginning right so third position second position fifth position sixth position like that we are talking about order in the sequence right so we must know what is the position of a one particular number in that sequence is that okay so hence nth term we call this as nth term nth term right nth term is given by this notation pn many literature talks about an where a is you know or you can always also see un all are same so nth term nth term nth term so n minus one-th term will be pn minus one this means element at the or the term at n minus one-th position position from beginning from beginning isn't it right so similarly tn minus two will be n minus 2th position n minus 2th term term rather term from beginning from beginning this is what we have studied so this is not position add this position this term so this is what we understand right so for example in this 3 7 11 15 19 23 let's say this is a sequence okay so let's say t4 is clearly 15 so t4 minus 1 will be 11 similarly t4 plus 1 will be t5 that is 19 and so on and so forth right so this is what is this notation is going to be used multiple number of times in fact all the formula or the derivation or the results will be based on nth terms so please be very very thorough with it so hence if you see the first term will be called so what will be the first term first term can be first term is an important thing for all these sequences so it can be written as a1 or t1 or u1 whichever depending on the case correct so you can name in any ways any of these ways I hope this part is clear to you in the next session what we are going to take take up is we need to understand how we define the nth term of a given sequence nth term of a given sequence since we talked about a well-defined law so can we convert nth term to a given formula which is based on few known things that will be the objective of the next session thanks