 Hello and welcome to the session let's work out the following problem it says find the common difference of an AP whose first term is 50 and the sum of whose first six terms is five times the sum of next six terms. So let's now move on to the solution. Now we are given that the first term of an AP is 50 let this be a we have to find the common difference so let the common difference be equal to d. Now we are given that the sum of first six terms of this AP is equal to five times the sum of the next six terms. So we have sum of first six terms that is a1 plus a2 plus a3 plus a4 plus a5 plus a6 is equal to five times the sum of next six terms that is a7 plus a8 plus a9 plus a10 plus a11 plus a12. Now we know that the nth term of an AP denoted by an is given by the formula a plus n minus 1 into d where a is the cost term n is the number of terms and d is the common difference. You must remember this formula and write it while using that same this as one. So from one we have a1 that is a the first term plus a2 which can be written as a plus 2 minus 1 d that is d plus a3 a3 is a plus 2 minus 3 minus 1 into d that is 2d plus a4 which is a plus 4 minus 1 into d that is 3d plus so on we can write a5 as a plus 4d and a6 as a plus 5d and this is equal to five times a7 that is a plus 7 minus 1d that is 60 a8 is a plus 7d a9 is a plus 8d a10 is a plus 9d a11 is a plus 10d a12 is a plus 11d. So we have a plus a plus a plus a6 times so this is 6 a plus d plus 2d is 3d 3d plus 3d is 60 60 plus 4d is 10d 10d plus 5d is 15d is equal to 5 times a plus a plus a 6 times 6 a plus 60 plus 7d is 13d 13d plus 8d is 21d 21d plus 9d is 30d 30d plus 10d is 40d 40d plus 11d is 51d so this implies 6 a plus 15d is equal to 30 a plus 255d so this implies 6 a minus 30 a is equal to 255d minus 15d so this implies minus 24 a is equal to 240d now we are given that the first term a is 50 so we have minus 24 into 50 is equal to 240 into d now d is equal to minus 24 into 50 upon 240 this implies d is equal to minus 24 into 50 upon 240 0 gets cancelled so we have d is equal to minus 5 hence the common difference is equal to minus 5 so this completes the question and the session bye for now take care have a good day