 Hello and welcome to the session. I am Asha and I am going to help you with the following question which says, in an AP, if P term is 1 upon Q and Q term is 1 upon P, prove that the sum of first PQ terms is half PQ plus 1 where P is not equal to Q. Let us now begin with the solution and we are given that the P term of AP is equal to 1 upon Q, the Q term is equal to 1 upon P. The P term is A plus P minus 1 into D, so this is equal to 1 upon Q where A is the first term of the AP and D is the common difference. Fine. And the Q term will be A plus Q minus 1 into D, so this is equal to 1 upon P. So let this be equation number 1 and this be equation number 2. So now, subtracting equation number 2 from 1, so first equation is A plus P minus 1 into D is equal to 1 upon Q and we have to subtract the second equation from it which is A plus Q minus 1 into D is equal to 1 upon P, so on subtracting. A cancels out with A and here we have P minus Q into D and on the right inside we have 1 upon Q minus 1 upon P which further implies that P minus Q into D is equal to P minus Q upon P Q, so D is equal to 1 upon P Q, let this be equation number 3. Now, substituting equal to 1 upon P Q in equation number 2 which is A plus Q minus 1 into D is 1 upon P Q is equal to 1 upon P or A plus 1 upon P minus 1 upon P Q is equal to 1 upon P which further implies that A is equal to 1 upon P minus 1 upon P plus 1 upon P Q which implies that A is equal to 1 upon P Q, so let this be equation number 4. Now, let us find the sum of first P Q terms, so this will be equal to P Q upon 2 into 2 into A plus P Q minus 1 into D, now substituting the values we have P Q upon 2 into A, so A is 1 upon P Q from equation number 4 plus P Q minus 1 into D is again 1 upon P Q and this is from equation number 3, so this is equal to P Q upon 2 to 2 upon P Q plus 1 minus 1 upon P Q which further implies P Q upon 2 is equal to 1 upon P Q plus 1 or we have P Q upon 2 into 1 upon P Q into 1 plus P Q inside the bracket, these two cancels out and we have half P Q plus 1, thus we are shown that sum of P Q terms is equal to half Q plus 1, so this completes the session, take care and have a good day.