 hey welcome friends once again on this session on solving problems related to arithmetic progression and specifically we are dealing with middle term problems okay so this question says in an AP with even number of terms prove that the sum of two middle terms is equal to the sum of first and the last terms okay so like we prove for the odd one we will try and prove for this one as well so let us assume so let's say let us say let us say that that there are there are n terms n terms in the AP okay and n is even n is even therefore we can say n is equal to 2k right for some non-negative k non-negative integer right non-negative means 0 1 2 3 4 yeah 0 1 2 3 4 integer non-negative integer k is a non-negative integer okay so middle term middle terms middle terms middle terms of an AP of an AP with even number of terms even number of terms right is given by r basically r p n by 2 and t n by 2 plus 1 isn't it we know that there are two middle terms is in case of even number of terms right so in this case what will happen n is 2k so this means they are pk and pk plus 1 these are the two terms two middle terms where n is 2k isn't it so dear friends pk will be from our basic definition of nth term it will be k minus 1d where t1 is equal to first term and d is what common difference no big deal correct common difference so hence similarly i can say tk plus 1 that is the second middle term is t1 plus k plus 1 minus 1 times d right this is first middle term and this is second middle term okay they are saying add of add both of them so let's add both of them so you'll get tk plus tk plus 1 will be equal to t1 plus t1 will give you 2t1 and this is simply d common k minus 1 and then this one and this one anyways will go so only k will be left d d isn't it right so hence this is nothing but 2t1 plus 2k minus 1d isn't it now remember what did we assume n is equal to 2k so we can right n is equal to 2k right now so hence this is and separate these t1 also so t1 plus t1 plus n minus 1d 2t1 was separated as t1 and t1 and this one now focus on this term this is nothing but the last term so this is t1 plus tn isn't it is equal to first term plus last term okay so some of the two middle terms we started from here is equal to first term plus last term that is what this question was asking you to prove right