 Hey friends, welcome again to the session on sequence and series and in this session We are going to take up some problems and we'll try to show you how to approach the problems Without having enough practice on all these problems. It will be difficult to comprehend or let's say Understand the concepts of sequence and series. So hence the emphasis should be more and more on problem solving So hence in the same coursework, you'll see lots of worksheets Recommend all of you. You can just download them and solve them. So few of them we will be solving as solved examples Just so that you can get an approach clear or understanding clear and then you can do rest of the problems yourself So here goes the first problem it says write the first three terms in each of the sequences defined by the following and Two sequences nth term are given one and two problems are there. Let us take You know them one by one. So here is what we are going to Do solution, right? So let's take one and a n is given as three n plus two very simple problem and They're asking to write three terms in each of the sequence. So first sequence is this now in our previous sessions We had taken nth term as tn So we had you know notified like that But we had also discussed that many literature talk about a n or u n or anything could be possible, right? So here they are talking about nth term and it is a linear term. We saw that you know in the previous sessions that such there are different types of Sequences and one type was wherein every nth term can be expressed as a linear polynomial. So this is what it is Okay, so a linear expression would be a better term because n is just an integer and is not or rather n is only positive integers and Is not even a rational number. Anyways, so n is always keep in mind and is n is a Positive integer, right? So I can write n is a positive Positive integer. You can't have n as a fraction or any rational number is out of question and is a Positive number, okay now So what is a1? You know how to do this replace the subscript or wherever you see n replace that by that number So 3 into 1 plus 2 so a1 is simply 5 No problem a2 will be simply 3 into 2 plus 2 so this will be 8 and a3 will be 3 into 3 plus 2 which is 11 right so if you extend this problem, let's say if they ask you find a 100 then what to do a 100 is simply 3 into 100 Plus 2 isn't it that is 3 not to 302 right so very easy to find out any nth term in this. Let's take up the next one so the second one says a n as n square plus 1 if you see this is nothing but a Quadratic expression right and we know that for quadratic what happens the second layer of difference becomes constant So let's generate few terms here and then we'll see that as well So clearly a1 will be simply deploy n is equal to 1 so 1 square plus 1 is 2 a2 will be 2 square plus 1 which is 5 What about a3? So if you see a3 is 3 square plus 1 that is 10 and A4 so we are just extending though only this much was required was for our analysis. We are going a little bit Ahead so this is 16 17 and then a5 is 5 square plus 1 26 now we have taken the sequence in our previous sessions if you remember now Let us find the difference of two consecutive terms so difference of 2 and 1 is 3 difference of 10 and 5 is 5 and difference of 17 and 10 is 7 and this difference is 9 so you can see the first level of difference is not constant Let's go for the second one. So this difference is 2 This difference is 2 which was but obvious and this one was also 2 Correct so level 1 difference is not constant It is in you know progression L 2 is constant and hence There is a quadratic expression here quadratic expression, right? So whenever the in any sequence the second level difference is constant You will use quadratic expression and if it is third level difference, which is constant then go for a Cubic expression. So this is what this is a fairly simple question You had to just deploy values of n and find out Different terms of the sequence and you can extend it to any nth term of the sequence