 Hello friends welcome to another session on sequence and series in the previous session. We were discussing about types of sequences and one type was Where the nth term is dependent on n itself direct function of n itself now we can have different types of these functions and We can see different types of sequences. So hence what could be the possible function on n? So when it is a linear form, let's say Okay, so one possibility is The fn is linear that is it is of the form of a n plus b where n is a positive integer and a and b a and b are real numbers Okay, and we are real numbers. This could be one type Another could be quadratic another could be quadratic right quadratic form where you see Fn could be of this form a n squared plus bn plus c Correct, then there is no limit to it c could be cubic where it could be a n's cube plus bn squared Plus cn plus d and so on and so forth all ABCD are real numbers, right? So hence you could have infinite such sequences So let's take At least, you know, we will be studying these three in little bit of more detail. So let's take up linear form linear form so Right, so let's say we have tn is equal to 3n plus 4 simple one just like that any a and b our choice 3 and 4 So what will be t1 t1 will be 3 plus 4 that is 7 t2 will be 3 into 2 Plus 4 that is 10 Isn't it 3 into 2 6 plus 4 10 t3 will be 3 into 3 plus 4 that is 13 and T4 is equal to 3 into 4 plus 4 that is 16 Now interesting thing is if you look at these Look at these two terms the common differences or there's a difference of three which is common to all the subsequent or successive successive Right successive terms. This is interesting, right? This is interesting. So is it only for In this case, no, so you can generalize it and see for example So you might question this that we have used a particular Function and hence it is happening. So hence, let's generalize it. So tn is equal to a n Plus b right, so what will be the next term tn plus 1 Will be a and in place of n I'll write a n plus 1 plus b And if you do this exercise tn plus 1 minus tn, what do we get we get a n plus Right plus b minus a n minus b you can check this out and this a n a n and b n b goes and hey ends a So a is a constant independent on independent of n okay, so always The successive or difference between two consecutive Terms will be a constant. That is what is the learning in case of linear form isn't it and We can find b also Right, so what is b if you if you really see b? B could be found out from this expression. So tn is equal to a n plus b Is it it so t1 will be equal to a plus b Isn't it so what will be b b is t1 minus a B is t1 minus a so the new relationship would be E n is equal to a times n plus even minus a Right. This is also a linear form. Let's inspect few more examples of this case So, let's say I have one three five seven nine Eleven this is clearly the difference between two consecutive term is a constant and which is two Okay, so clearly the constant difference difference was If you see this tn minus tn plus one minus tn was a so a we will get very easily So a is what a is any tn plus one minus tn So that means difference of any two consecutive term, which is anyways constant in all the cases So hence a will be simply two and what will be b? We saw b as t1 minus A here t1 minus a what is t1 first term first term is one and a was two So one minus two that is minus one So dear friends in this case nth term will be given by a n plus b Isn't it which is now a is two so two and minus one that will be the Generic term for this. So what did we do in this exercise? We figured out We figured out the Expression given the sequence you can find out the nth term, right? So let's take an exact question on this. So let's say I have five nine 13 17 21 25 this is a sequence and what type of sequence it is the consecutive term is constant for difference is constant for Four right so all our four. So what is this form? This form is a linear form because in linear form We just learned that the tn can be expressed as a n plus b in linear form the consecutive terms will have Common difference so tn is equal to a n plus b and we also learned a was equal to nothing But the common difference let me call it CD so common difference in this case is four and B was nothing but first term minus the common difference a Which is nothing but five minus four, which is one So now if we write the formula, I will get tn is equal to Four n plus one four n plus one this should be my formula for each of the term So let's check whether it is true or not. So let's say t3 is third term third term will be four into three Plus one it should be 13 and let's check 30 third term. Yes indeed. It is 13 What about p6 p6 should be four into six plus one which is 25 and yes, it is 25 here see one two three four five and six term is 25 so it matches right so now someone asked me give me a hundredth term in this sequence So I don't need to break my head on to this. I will simply apply this formula and Say it is four not one Okay, so that's how We can if a sequence is given who's constant or the it's linear type Sequence that is the first layer of difference is constant Then the sequence can have an expression in linear form Every term can have an expression in linear form You have to just find out a and b from the given data And once you have found out you can generalize the nth term and you can find out any Tn in that sequence