 Hello and welcome to the session. In this session we discussed the following question which says find the nth term and deduce the sum to n terms of the series 1 into 3 square plus 2 into 5 square plus 3 into 7 square plus and so on. Before moving on to the solution, let's discuss one very important theorem according to which we have that if the nth term of the series be given by Tn is equal to say a n cube plus b n square plus cn plus d then we have the sum of the series to n terms given by Sn is equal to a into summation n cube plus b into summation n square plus c into summation n plus b into n. Then next we have summation n is equal to n into n plus 1 d whole and this whole upon 2 then summation n square is equal to n into n plus 1 d whole and this whole into 2 n plus 1 d whole and this whole upon 6 summation n cube is equal to n into n plus 1 d whole and this whole upon 2 and this whole square or you can also say summation n whole square since we know that summation n is n into n plus 1 d whole and this whole upon 2. This is the key idea that we use for this question. Let's now proceed with the solution the given series is 1 into 3 square plus 2 into 5 square plus 3 into 7 square plus and so on. We are supposed to find the nth term of the series and also the sum of the n terms of the series. Let's now find out the nth term of the series. So the nth term of the series given by tn would be equal to nth term of the series 1, 2, 3 and so on and this whole into nth term of the series 3 square 5 square 7 square. Now the nth term of the series 1, 2, 3 and so on would be n and the nth term of the series 3 square 5 square 7 square and so on would be 2n plus 1 d whole square. So this is tn which could also be written as n into 4n square plus 4n plus 1 d whole or this is equal to 4nq plus 4n square plus n that is tn. So nth term of the series is given by tn and this is equal to 4nq plus 4n square plus n. Now we know that if we are given tn equal to anq plus bn square plus cn plus d then sum of the series to the n terms given by sn would be equal to a into summation nq plus b into summation n square plus c into summation n plus dn. So using this we can say that the sum of the series given by sn would be equal to 4 into summation nq plus 4 into summation n square plus summation n. We know that summation nq is equal to n into n plus 1 d whole whole upon 2 and this d whole square. So putting this here we get sn is equal to 4 into n into n plus 1 d whole this whole upon 2 and this whole square plus 4 into summation n square and summation n square is equal to n into n plus 1 d whole into 2n plus 1 d whole and this whole upon 6. So here we have 4 into n into n plus 1 d whole this whole into 2n plus 1 d whole and this whole upon 6 plus summation n which is equal to n into n plus 1 d whole and this whole upon 2. So putting this we get n into n plus 1 d whole and this whole upon 2. Now here 2 2 times is 4 and 2 3 times is 6. So we get sn is equal to 4 upon 4 that is 2 square is 4 into n square into n plus 1 d whole square plus 2 upon 3 into n into n plus 1 d whole into 2n plus 1 d whole plus n into n plus 1 d whole and this whole upon 2 this 4 cancels with 4 and now we take n into n plus 1 d whole common and here we get n into n plus 1 d whole that is from here when we take n into n plus 1 d whole common we are left with n into n plus 1 d whole plus 2 upon 3 into 2n plus 1 d whole that is we have taken this common plus 1 upon 2 since we are taking n into n plus 1 d whole common. So this is further equal to n square plus n the whole into n into n plus 1 becomes n square plus n plus 4n upon 3 plus 2 upon 3 plus 1 upon 2 d whole this is equal to n square plus n the whole and this whole into now taking LCM here and the LCM would be 6 6 into n square is 6n square plus 6 into n is 6n plus 3 2 times is 6 and 2 into 4n becomes 8n plus 3 2 times is 6 and 2 into 2 is 4 plus 2 3 times is 6 and 3 into 1 becomes 3. So further we get n square plus n the whole into 6n square plus 14n that is 8n plus 6n is 14n plus 4 plus 3 is 7 and this whole upon 6. So this would be equal to now multiplying these two we get 6n square into n square becomes 6 into n to the power 4 plus n square into 14n is 14n cube plus 7 into n square is 7n square plus 6n square into n is 6n cube plus 14n into n is 14n square plus 7 into n is 7n and this whole upon 6. So this is equal to 6 into n to the power 4 plus 20n cube plus 21n square plus 7n and this whole upon 6 this is Sn. So thus we get the sum of the n terms of the series as this. So finally we can say that the nth term of the series is equal to this that is 4n cube plus 4n square plus n 4n cube plus 4n square plus n and the sum of the n terms of the series is equal to 6 into n to the power 4 plus 20n cube plus 21n square plus 7n and this whole upon 6. So this is our final answer this completes the session hope you have understood the solution of this question.