 Hi and welcome to the session. I am Arsha and I am going to help you with the following question which says if S1, S2 and S3 are the sum of first 10 natural numbers, their squares and their cubes respectively show that 9S2 square is equal to S3 into 1 plus 8S1. Let us now begin with the solution and S1 is equal to the sum of natural numbers that is 1 plus 2 plus 3 plus 2 up to n and the sum of first 10 natural numbers is given by n into n plus 1.2 and S2 is the sum of squares of the first 10 natural numbers so that means S2 is equal to 1 square plus 2 square plus 3 square up to n square. Now the sum of squares of first 10 natural numbers is n into n plus 1 into 2n plus 1 upon 6. S3 is equal to the sum of cubes of first 10 natural numbers so we have 1 cube plus 2 cube plus so on up to n cube and this is equal to n plus 1 whole square upon 4 and we have to show that 9S2 square is equal to S3 into 1 plus 8S1 so let us begin with the right hand side and we will show that it is equal to the left hand side. So the right hand side is S3 into 1 plus 8S1 now S3 is n square into n plus 1 whole square upon 4 into inside the bracket here 1 plus 8 times of S1 which is n into n plus 1 upon 2 which is already written as n square into n plus 1 whole square upon 4 into 1 plus 4n into n plus 1 this is equal to n square into n plus 1 whole square upon 4 into 1 plus 4n square plus 4n. Now let us sacrifice this bracket so we have 4n square plus 4n plus 1 which can forever written as 4n square plus 2n plus 2n plus 1. Now taking 2n common from the first two terms and one common from the last two terms we have 2n into 2n plus 1 plus 1 times 2n plus 1 so this is equal to 2n plus 1 into 2n plus 1 plus 2n. Therefore this can forever written as n square into n plus 1 whole square upon 4 into now simplifying this bracket it gives us that 4n square plus 4n plus 1 is equal to 2n plus 1 whole square fine. So this is further equal to n into n plus 1 into 2n plus 1 into 2n plus 1 whole square upon 4. Now multiplying the numerator and denominator by 9 we have 9 times n into n plus 1 into 2n plus 1 whole square upon 36 which can forever written as 9 times of n into n plus 1 into 2n plus 1 upon 6 whole square and this is equal to 9 times of s2 since this is the sum of squares of first and natural numbers and then we have square and therefore we have proved that 9s2 square is equal to s3 into 1 plus 8s1. So this is the left hand side and this completes the solution take care and have a good day.