 Hi and welcome to the session. I am Asha and I am going to help you with the following question which says, show that 1 into 2 square plus 2 into 3 square plus n into n plus 1 whole square is equal to 1 square into 2 plus 2 square into 3 plus so on up to n square into n plus 1 is equal to 3 n plus 5 upon 3 n plus 1. Let us now start with the solution and the left hand side is 1 into 2 square plus 2 into 3 square plus so on up to n into n plus 1 whole square upon the denominator is 1 square into 2 plus 2 square into 3 plus so on up to n square into n plus 1 which can further be written as summation k running from 1 to n k into k plus 1 whole square upon summation k square into k plus 1 k running from 1 to n. Now this is further equal to summation k into k square plus 2 k plus 1 upon summation k 2 plus k square k running from 1 to n. This is further equal to summation k cube k running from 1 to n plus 2 times summation k running from 1 to n k square plus summation k running from 1 to n upon summation k cube k running from 1 to n plus summation k square k running from 1 to n. This is further equal to n into n plus 1 upon 2 whole square plus twice n into n plus 1 into 2 n plus 1 upon 6 plus n into n plus 1 upon 2 and the denominator n into n plus 1 upon 2 whole square plus n into n plus 1 into 2 n plus 1 upon 6. This is further equal to taking n into n plus 1 upon 2 common here we have n square plus n upon 2 plus 2 times of 2 n plus 1 upon 3 plus 1 and the denominator we have n into n plus 1 upon 2 again taking common here we have n square plus n upon 2 plus 2 n plus 1 upon 3 which is further equal to these two answers out and here we have LCM as 6 here we have 3 n square plus 3 n plus 2 to the 4 and 4 to the 8 n plus 4 ones of 4 plus 6 upon the denominator we have again LCM as 6 and then the numerator 3 into 3 n square plus 3 n plus twice of 2 n plus 1 that is 4 n plus 2 which is further equal to 3 n square plus 11 n plus 10 upon 6 into 6 upon 3 n square plus 7 n plus 2. Now let us factorize first 3 n square plus 11 n plus 10 so this can be written as n plus 3 n square plus 5 n plus 6 n plus 10. Now taking n common from the first two terms and 2 common from the last two terms it can further written as n into 3 n plus 5 plus 2 times 3 n plus 5 so this is equal to n plus 2 into 3 n plus 5. Now let us factorize the denominator which is 3 n square plus 7 n plus 2 so this can be written as 3 n square plus 6 n plus n plus 2. Now taking 3 n common from the first two terms and 1 common from the last two terms this can further written as 3 n into n plus 2 plus 1 times of n plus 2 which is further equal to 3 n plus 1 into n plus 2 and therefore with the help of these two factorize equation the LHS can further written as 6 cancels off with 6 and 3 n is square plus 11 n plus 10 is equal to n plus 2 into 3 n plus 5 and 3 n square plus 7 n plus 2 can be written as 3 n plus 1 into n plus 2. Now canceling the common factor such as n plus 2 we have 3 n plus 5 upon 3 n plus 1 which is the right hand side of what we have to prove. Therefore LHS is equal to RHS or we can say that 1 into 2 square plus 2 into 3 square plus so on up to n into n plus 1 whole square upon 1 square into 2 plus 2 square into 3 plus so on up to n square into n plus 1 is equal to 3 n plus 5 upon 3 n plus 1. So this completes the solution take care and have a good day.