 Hello students let's work out the following problem it says the 8th term of an arithmetic progression that is AP is 37 and 12th time is 57 find the AP. So let's now move on to the solution. Now we know that the nth term of an arithmetic progression that is AP is denoted by AN and is given by the formula A plus n minus 1 into D where A is the first term of the arithmetic progression and D is the common difference. So you need to remember this formula because we will be using this formula. Now we are given the 8th term and the 12th term. Now 8th term is A8, A8 is 37 which is equal to A plus n is 8 n minus 1 into D that is 37 is equal to A plus 7 D similarly A12 which is equal to 57 is equal to A plus 12 minus 1 into D. So this implies 57 is equal to A plus 11 D let's name this as 1 this is 2. Now we'll solve equation 1 and 2 for A and D so we subtract equation 2 from 1 so we have equation 1 is A plus 7 D is equal to 37 equation 2 is A plus 11 D is equal to 57 now since we are subtracting sign will change so plus A gets cancelled with minus A 11 D 7 D minus 11 D is minus 4 D 37 minus 57 is minus 20 so this implies D is equal to minus 20 upon minus 4 that is equal to 5 so the common difference is 5. Now we can easily find out the value of A by putting x is D is equal to 5 in any one of the 2 equations in 1 so we have A plus 7 into 5 is equal to 37 so this implies A plus 35 is equal to 37 and this implies A is equal to 37 minus 35 so this implies A is equal to 2 so we have D is equal to 5 and A is equal to 2 now AP is given by the formula A A plus D A plus 2 D A plus 3 D A plus 4 D and so on now A is 2 A plus D that is 2 plus 5 A plus 2 into D which is 5 2 plus 3 into 5 2 plus 4 into 5 so this becomes 2 7 2 plus 10 is 12 2 plus 15 is 17 and the next is 22 and so on so this is AP that is arithmetic progression is 2 7 12 17 22 and if you see that here in this AP if you keep on adding 5 to the previous number you will get the 8 term as 37 and 12 term as 57 so this is the required AP so this completes the question do remember the formula for the nth term of an AP and you also need to remember how do we write the arithmetic progression so this completes the question and the session bye for now take care have a good day.