 Hello students, let's work out the following problem. It says find the sum of all two digit numbers, it's natural numbers which are divisible by 4. So let's now move on to the solution and the two digits natural number which are divisible by 4, 20, 24, 28, 32 and so on till 96. Right, we have to find the sum of all these numbers. Now we say that this is an arithmetic progression where the first term is 12 and the common difference is 4. Right, because if you subtract first term from the second term you will get the difference as 4 and if you subtract second term from the third term you will get the difference as 4. So the common difference is 4 and we know that the sum numbers of an AP that is arithmetic progression is given by the formula Sn is equal to N by 2 into 2A plus N minus 1 into D. You need to remember this formula and you have to write this formula and here A is the first term of the arithmetic progression, D is the common difference is the number of terms. So to find the sum of these numbers which are divisible by 4 we need to find the number of terms in this arithmetic progression. Now we know that the nx term of this AP is 96 and the nx term in AP that is arithmetic progression is given by An is equal to A plus N minus 1 into D. You also need to remember this formula. Now nx term is 96. So 96 is equal to A plus N minus 1 into D. Now here A the first term is 12 and the common difference is 4. So it is 12 plus N minus 1 into 4 to write A is 12 and D is 4. So this implies 4 into N minus 1 is equal to 96 minus 12 and this implies 4 into N minus 1 is equal to 84 and this implies 4N minus 4 is equal to 84 and this again implies 4N is equal to 84 plus 4 and this implies 4N is equal to 88 and this implies N is equal to 88 upon 4 and this implies N is equal to 22. So the number of terms in this AP are 22. Now using this formula we can find out the sum of 22 terms. Let's name this as A. So put N is equal to 22, A is equal to 12 and D is equal to 4 in A. So we have SN is equal to NY2 that is 22 by 2 into 2A that is 2 into 12 plus N minus 1 into D. Now N is 22 minus 1 into 4. So this is equal to 11 into 24 plus 21 into 4, 22 minus 1 is 20, 22 minus 1 is 21 into 4 and this is again equal to 11 into 24 plus 84 and this is equal to 11 into 108 and 11 into 108 is 1188 and this is the required sum is equal to 1188. So this completes the question and the session. Do remember all the formulae and write all the formulae and do take care of all your calculations. So bye for now. Take care. Have a good day.