 welcome friends to another session on sequence and series and in this session we are going to discuss middle term of a finite AP okay so we know or we have learned about an AP that is arithmetic progression and let's say there are n terms n terms of an AP are there okay often arithmetic progression are there let us assume that right and what are the terms then P1, P2, P3, P4 and so on and so forth let's say Tn-2 Tn-1 and finally Tn how many terms these are n terms of an AP okay and I'm interested in finding out the middle term middle term as in exactly the term which is lying exactly at the midpoint of the entire sequence over here so how to find out let's take some examples and understand let's say there are let's say let's say n is equal to 100 okay so hundred terms are there so which term do you think is the middle term now obviously this is a you know difficult task why because if you see if you say that 50th term let's say 50th term is the middle term then there is a problem why because if there are let's say this is 50th term right so on this side there are 49 terms is it it from 1 to 49 and on this side there are 50 terms so this is not correct that 50th term is the middle term isn't it 50th this side 50 terms why because from 51 52 if you count till let's say 100 okay so n was 100 so how many are there 50 are there 50 terms are there but this side from 1 to 2 to 3 let's say till 49 49 right so here 49 terms are there so they are not equal right so we see that there is a problem but if n is 101 let's say n is 101 then what happens 50 first term becomes the middle term very easily right 51st term why because now there will be 50 terms on this side from 1 to 50 and here also there will be 50 terms from 52 to 101 right so these are 50 terms each from 52 so you count 52 to 101 will be 50 terms right so so that is you know balanced so if so hence we see two cases are there when case one case one when case one is when let's say n is even then clearly there is no one middle term so the way out which we have found out is if n is even there will be there are two midterms there are two middle terms now right why two terms so on the right hand side and left hand side of these two middle terms in the sequence the number of terms are same right so in this case if you go back to this case right let's say in case of n is equal to 100 so you can see that if I treat 50th and 51st these are the midterms middle terms so in this side also there will be 49 terms 49 terms from 52 to 100 and this side also there are 49 terms okay so in a way these two will be the middle term so hence how do we generalize so we say that it is n by 2 n by 2th term is the middle term and n by 2 plus 1 plus 1 1th term right these two are the middle terms in this case if n is 100 then t 100 by 2 and t 100 by 2 plus 1 are the middle terms are middle terms right is it it and if n is let's say another another even number let's take n is 80 okay so in this case t 80 by 2 that is t 40 and t 80 by 2 plus 1 that is t 41 so t 40 and t 41 are middle terms so two middle terms in case of n is equal to even so let's take the case when k the n is odd right so n is odd it was very easy for us because if you take the example n is 100 so middle term was sorry n cannot be 100 when n is odd so n is 101 say so middle term middle term was how much or which term was middle term middle term was simply 51 right t 51 was the middle term and how did we calculate so if you see this is nothing but e of 100 or t of 101 101 then add 1 to it and divide by 2 right so t this is the position number so you can write like that t 101 plus 1 by 2 that is t 51 so generalizing we will get t n plus 1 by 2 is middle term right or n plus 1 by 2 th term n plus 1 of n plus 1 by 2 th term is the middle term right is the middle term so in case of even number there are two so just summary summary of the discussion if n is equal to even then two midterms two middle terms exist is remember which all t n by 2 and t n by 2 plus 1 okay and if n is equal to an odd number then one middle term exists one middle term exists at what location t n plus 1 by 2 right so keep this in mind