 Hello friends. So in the previous session we found out the general nth term from the beginning of the AP in this Session we are going to take up nth term from end of an AP, right? So there are two ends beginning and you know, there will be end of the AP So from the one side you can count from the other side also you can count So what is nth term from end of an AP? So let's observe the trend. Okay, so for that What we'll do is we'll first draw a small table Right, and here we'll write from beginning. So from B, let's say and from end so E Okay, and let's try to understand the positions Okay, so I hope this is clear. So let's say from beginning Position is first and here let's say and before that we have to also observe that there are how many terms M terms in an AP so something which is first You know from beginning is ameth from the The other other side, isn't it? Ameth position. I hope this is clear to you. So So let me start with yeah, so from beginning if it is first from end it will be M Ameth position if it is second it will be M minus 1 If it is third term from the beginning it is M minus second term from the correct From the last right. This is how so hence if it is ameth here Ameth term from the beginning will be first term from the end Isn't it? So let's reverse this process. Now. I'm saying from end here And this is from B. What do I mean? Okay, so let's say from end Okay, now I'm I'm taking from and first term. I'm just reversing it So from and first term is from beginning ameth term. So this will be called P M right from End second term is M Minus one right so the value of index will be M minus one So let me write it properly. So the index is going to be M minus one and hence term is P M Minus one, please understand these terms are when I write P and suffix M and M minus one these are from beginning So this is M minus one of them from beginning Okay, now when from end third term is M minus two F term from the beginning, right? So P M minus two P M minus two is The third term from the end, right? So from end nth term All right from end if I have to find out nth term if you observe it will be simply. So this will be N Minus N Plus one right the index will become M minus and plus one you can observe that So what a two corresponds to M minus two minus one Correct three corresponds to M minus one lesser than three, right? So hence it is actually I have to write this Like M minus. So actually I should have written it M minus N minus one like that Correct. So three corresponds to M minus two. So N will correspond to M minus N minus one I hope this is clear Correct. So look closely. You will get it. So second term from end is M minus one from beginning Third term from end is M minus two from beginning. So nth term from end will be M minus N minus one of term. So hence this comes out to be M minus N plus one and hence I can write this as T M minus N plus one, right? So nth term from beginning. So what is the conclusion? nth, oh sorry nth term from end term from end is equal to M minus N plus one of term from beginning Correct. This is what we learned. So nth term from when M minus one and M minus one plus one of term from beginning can be written as T one. This formula we already have found out M minus N plus one minus one D Correct. Why minus one? Because generic term of an AP is TN is equal to first term plus N minus one D. So M minus one plus one of term will be T one plus M minus one plus one minus one D, right? So that is equal to T one plus M minus N D, right? So nth term from end will be T one plus M minus one D T one plus M minus N D where M and M is how much? M is number of terms number of terms in that AP terms in AP Correct. If we want to express it in the form of or from the last term itself, for example What do I mean? So here if you see there is the first term mentioned the first term instead of that If I want to take it from the last term, so don't you think that P M minus one is nothing but P M minus D one D less than the last term So T M minus two, right? So M minus 2th term from the beginning is P M last term minus 2 D Right. So hence You know nth term from the end so nth term from the end will be yeah, so Let's say if you want to know nth term from end from end What will be nth term from end? So first term, this is this this guy is second term from end This guy is third term from end So nth term from end will be simply T M Minus n plus one see the same thing. I don't get confused. So T now this is T M minus n plus one from the You know, so the index will be M minus n plus one and this will be how much T M minus M minus n M minus n No, not M minus n it will be simply so you can use the correspondence So if when it is one so here is minus one when it is minus two here, it is minus two So when it is n minus one here, so it will be n minus one D. I Hope this is clear. So don't get confused. This is T actually M minus n minus one this one And whatever was here is coming with D there. So this one is equal to T M minus n minus one D Okay, so M minus n minus one of yep So nth term from end is equal to last term. So I can write like that nth term From end is equal to last term Minus n minus one into common difference Right, this is what We can find out the nth term from both the beginning and the end. So just to summarize E N from beginning, right? So whenever I write E N sub subscript N means it is from beginning is nothing but E one plus n minus one D Okay, and nth term from From the end will be T M minus n plus one. We just learned this is nth term from Beginning right, this is nth term From end Right from nth term from end is as good as saying as T M minus n plus one and we learned this one as T one plus M minus and D and if you want to express it from the Last term then this is T M minus M. Sorry not M N and Minus one D, right? So T N is nth term from the beginning and nth term from end is T M minus n plus one, right? So nth term from beginning nth term from end. So we have to express as T and subscript the moment I write E and subscript something it always say it is from beginning So hence don't get confused. So nth term from E is M minus n plus one of term from B from beginning right and that has two expressions in terms of first term and in terms of last