 Hi and welcome to the session. I am Asha and I shall be looking here with the following question which says if the sum of first p terms of an AP is equal to the sum of first q terms then find the sum of first p plus q terms. Let us now start with the solution and here we are given that sum of first p terms of AP is equal to the sum of first q terms that is p upon 2 into 2a plus p minus 1 into d is equal to q upon 2 into 2a plus q minus 1 into d or this equation can further be written as p upon 2 into 2a plus p minus 1 into d minus 2 upon 2 into 2a plus q minus 1 into d is equal to 0 or we have 2a upon 2 common in the bracket p minus q plus again taking d common and we have p minus 1 into p upon 2 minus q minus 1 into q upon 2 is equal to 0. So, we have a into p minus q plus d times of p into p minus 1 minus q into q minus 1 upon 2 is equal to 0 or this can further be written as a into p minus q plus d p square minus p minus q square plus q upon 2 is equal to 0 or a into p minus q plus d now p square minus q square is p minus q into p plus q minus of p minus q upon 2 is equal to 0 or we have a into p minus q plus d into p minus q taking common we have p plus q minus 1 is equal to 0. So, now taking p minus q common we have p minus q into a plus p plus q minus 1 upon 2 into d is equal to 0 or p minus q into 2a plus p plus q minus 1 into d is equal to 0 multiplied by 2 since 2 is the LCM. Now since we are given that p is not equal to q. So, this implies p minus q is not equal to 0 since we are given that the sum of first three terms of any p is equal to the sum of first few terms therefore p is not equal to q. Now since this is not equal to 0 this will imply that 2a plus p plus q minus 1 into d is equal to 0. Now let us find the sum of first few terms whose formula is p plus q upon 2 into 2 into a plus p plus q minus 1 into d. This is further equal to p plus q upon 2 into the value of the square bracket is 0 just we have find the both. So, this is equal to 0 and anything multiplied by 0 is 0 therefore our answer is just q terms equal to 0. So, this completes the session take care and have a good day.