 Hi, this is Asha explaining this portion of your book which says the sum of first four terms of an AP is 56. The sum of last four terms is 112. If the first term is 11, find the number of terms. Let's now start with the solution and we are given that the first term Let us denote it by small a is equal to 11 and the sum of first four terms is equal to 56 So let the first four terms be a plus d a plus 2d and a plus 3d where D is the common difference Now we are given that the sum of these four terms is equal to 56. So we have a a plus d plus a plus 2d plus a plus 3d is equal to 56 or if we have 4a plus 6d is equal to 56 Now a is 11 so on substituting 11 we have 4 into 11 plus 6d is equal to 56 or we can say that 6d is equal to 56 minus 44 So 6d is equal to 12 or d is equal to 2 So our common difference is 2 let this be equation number 1 Now also we are given that sum of last four terms equal to 112 Select the four terms be a n plus a n minus 1 plus a n minus 2 Plus a n minus 3 from the last so their sum is equal to 112 Now a n is a plus n minus 1 into d a n minus 1 is a plus n minus 2 into d a n minus 2 is a plus n minus 3 into d and a n minus 3 is a plus n minus 4 into d So this is equal to 112 or we can say that 4a plus 4nd Minus 10d is equal to 112. Now substituting the values of a and d we have 4 into 11 plus 4n into 2 minus 10 into 2 is equal to 112 or we have 44 plus a 10 minus 20 is equal to 112 or 8n is equal to 112 minus 24 Which implies that a 10 is equal to 88 or n is equal to 88 upon 8 which is equal to 11 And that's the number of terms is equal to 11 So this completes the solution take care and have a good day