 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 the first term is true and the sum of first five terms is one fourth the sum of next five terms. Show that the twentieth term is minus 112. Let us now begin with the solution. Let D be the common difference. First five terms be equal to 5 upon 2 plus 5 minus 1 into D where A is the first term and D is the common difference. This further equal to 2 upon 2 into, the first term is 2, so 2 into 2 plus 4 into D. 5 upon 2 into 2 into 2 plus 2D into 5 into 2 plus 2D. Thus the sum of five terms is equal to 5 into 2 plus 2D. Now we have to solve the next five terms. That is the sixth term, seventh term, eighth term, ninth term and tenth term. So we have A plus 5D which is the sixth term plus A plus 6D which is the seventh term plus goes on up to A plus 9D which is the tenth term. Now this is equal to total number of terms are 5 upon 2 into 2 into A. So the first term of the sequence is A plus 5D plus N minus 1, so 5 minus 1 into D and the common difference is D. So this is equal to 5 upon 2 into 2 into A plus 5D plus 4D which is equal to 5A plus 5D plus 2D which is further equal to 5 into A plus 7D. And therefore next five terms is equal to 5 into A plus 7D. Now as we are given that sum of five terms is equal to 1 by 4 into sum of next five terms, five terms of 5 into 2 plus 2D is equal to 1 upon 4 into sum of next five terms is 5 times of A plus 7D or we have 5, 4, and 20 on cross multiplying 2 plus 2D is equal to 5 and the first term of the sequence is 2. So we have 2 plus 7D or we have 22 is a 40 plus 40D is equal to 10 plus 4D plus 35D. How we can say that 40D minus 35D is equal to 10 minus 40 or 5D is equal to minus 30 or D is equal to minus 6. So the value of D is minus 6. Now we will show that the 20th term let us denote it by A20 is equal to minus 100 and 12. Now A20 which is the 20th term of the AP is given by A plus 20 minus 1 into D. Now A is plus 20 minus 1 is 19 into D is minus 6. This is further equal to 2 plus minus 114 which is further equal to minus 112 and this the 20th term is equal to minus 112. So this completes the session. Take care and have a good day.