 Welcome friends once again to this session on sequence and series and today we are going to take up general term of NAP. In the previous session you saw what an arithmetic progression was and we also took some illustrations we explained through examples what an arithmetic progression look like. Now we are going to define the general term of NAP what do I mean? So let's say there is an arithmetic progression given arithmetic progression whose short form I am writing as an AP and let's say these are the terms there T1 and T2, T3, T4 so on and so forth and let's say there are N terms TN okay there are these terms in the AP right. So as you know from your previous session we have learned this is the first term of deep progression this is the second term and so on and so forth and this one is nth term nth term right so as we have been studying that arithmetic progression is governed by a particular formula which actually describes any nth term so we are interested in knowing that formula. Now what is the characteristic of an arithmetic which we learned in the previous session that any two consecutive terms if you take the difference is same isn't it this is what describes an AP correct this is D common difference it's called so T4 and T3 is D so that's what we learned right and so on this is how right so TN minus 1 and TN these are the terms and everywhere the consecutive terms will have a difference of D this is how the AP has been defined isn't it so my dear friends can I not say that T2 so we we have got this relationship T2 minus T1 is D and similarly T3 minus T2 is D correct and T4 minus T3 is D and so on and so forth isn't it so you keep on going like that so TN minus TN minus 1 is D right this is what we have learned okay now so from this expression or equation I can write T2 is equal to simply T1 plus D okay and from here T3 can be written as T2 plus D isn't it and what is T2 guys in the previous equation if you see T2 is T1 plus D from here so can I not write that T2 as T1 plus D plus D right so it is T1 plus 2D right let's evaluate T4 so if you see T4 is T3 plus D how do I know this from here right so T4 is T3 plus D but what is T3 T3 was T1 plus 2D so this is T3 plus D so hence this becomes T1 plus 3D is it so what do I learn you know if you can carry on like that so you'll get T5 as T1 plus 4D you can check this out T6 as T1 plus 5D and so on and so forth so can we generalize it so if you see if you want to find out Tn so Tn will be simply first term plus n minus 1D how do I know that this is n minus 1 so if you look closely when this term here is 4 here the number is 4 minus 1 right when this one is 5 here with D the coefficient is 5 minus 1 here when it is 6 this is 6 minus 1 so hence if it is n this will be n minus 1 I hope you understood this right so hence any nth term right this is nth term we say nth term nth term means the term at the nth position correct similarly if you know so let's take some examples of our example if n is equal to 10 then 10th term 10th term of the AP will be what 10th term T10 will be simply T1 plus 9D right n is equal to let's say 15 right so you want 15th term so 15th term 15th term of that AP will be T15 and this will be nothing but first term plus T minus 1 that is 14 n minus 1 14D correct so this is how you can find any so for example in terms of variables also let's say if I want mth term of the AP mth right term at the mth position so you have Tm will be equal to T1 plus m minus 1D let's say you want mnth position right so let's say m is some value n is some value so mnth position so if you want to find out mnth term so this will be simply Tmn so is equal to T1 plus mn minus 1D these are our illustrations what does this last thing mean many people get confused here so let's say m is equal to 3 and n is equal to 2 so mn will be equal to 6 so mnth term will be 6th term right so mnth term is 6th term right so in that case what will happen Tmn is T1 first term plus m into n minus 1D right so in this case T6 or T3 and m is 3 n is 2 is T1 plus 6 minus 1D correct right so for that matter you can find out any you know any term from the beginning now if you are yes just to you know make it very clear that these are these are nth terms right n is a generic number nth term from beginning from beginning right right and it's term from the beginning of which AP T1 T2 T3 T4 like that that's a Tn right this is right and there are you know in this case n is considered to be the you know let's say there are n or m terms in the AP so let's say n plus 1 and so on and so forth let's say there are m terms m m m from m for mango m terms right so T1 T2 T3 T4 Tn Tn plus 1n dot dot dot till TM these are the terms of NAP correct so nth term from beginning mth nth term from the beginning is Tm is how much first term plus n minus 1D correct is it now the same term from the last term so what is the last term guys TM correct if I count from here how do I found how do I find nth term from here right so let's say this is Tn this Tn expressed in form of TM how do I get it is the next agenda okay