 Hi and welcome to the session. I am Asha and I am going to help you with the following question which says Find the sum to n terms of each of the series 1 square plus 1 square plus 2 square plus 1 square plus 2 square plus 3 square plus so on. So let us now begin with the solution. And let S be the given series so S is equal to minus square plus 1 square plus 2 square plus 1 square plus 2 square plus 3 square plus so on. So let the kth term of the series be ak so ak will be equal to 1 square plus 2 square plus goes on up to k square that is summation m running from 1 to k m square and its formula is k into k plus 1 into 2k plus 1 upon 6 so ak is equal to k square plus k into 2k plus 1 upon 6. Now we have to find the sum to n terms so taking summation on both the sides we have summation ak k running from 1 to n 1 upon 6 summation k running from 1 to n. Now solving these two brackets we have 2k cube plus k square plus 2k square plus k which is further equal to 1 upon 6 summation k running from 1 to n 2k cube plus 3k square plus k. This is further equal to 1 6 summation k running from 1 to n k cube plus 3 upon 6 summation k running from 1 to n k square plus 1 upon 6 summation k running from 1 to n k. This is further equal to 2 upon 6 and here we have n square into n plus 1 whole square upon 4 plus 3 upon 6 into n into n plus 1 into 2n plus 1 upon 6 plus 1 upon 6 into n into n plus 1 upon 2. Now 2 to the 4, 3 to the 6 and now let us take n into n plus 1 upon 12 common suppose we have n into n plus 1 plus 2n plus 1 plus 1 which is equal to n into n plus 1 upon 12 into n square plus 3n plus 2 twice this bracket here n square plus 3n plus 2 which was written in the middle term n will be written as n square plus 2n plus n plus 2. Now taking n common from the first two terms and one common from the last two terms it can further written as n into n plus 2 plus 1 into n plus 2 which is further equal to n plus 2 into n plus 1. So we have n into n plus 1 upon 12 into n plus 1 into plus 2 that is n into n plus 1 whole square into n plus 2 upon 12. Therefore this is given by into n plus 1 square into n plus 2 upon 12. So this is the sum to n terms whose series is given to us. So this completes the session. Thank you and have a good day.