 Hello and welcome to the session. In this session we discuss the following question which says, find the sum of the series 2 plus 5x plus 8x square plus 11x cube plus m star n to n pounds. Hence write down the sum to infinity with x less than 1 numerically. First of all let's see what is a mathematical geometric series. The general or standard sum of such a series is given by a plus 3d whole into r plus a plus 2d whole into r square plus a plus 3d whole into r cube into d and this whole into r to the power n minus 1 plus and so on. A mathematical geometric series found by multiplying the corresponding terms of an a p and a gp. And in this case the a p is a plus a plus d plus a plus 2d plus 3d minus 1 into d plus the gp that is a geometric progression is 1 plus r plus r square plus r cube plus minus 1 plus key idea that we use in this question. Let's now move on to the solution. We are supposed to find the sum of this series to n terms to write down the sum to infinity of the series with x less than 1 numerically. Now the given series which is 2 plus 7 plus and so on e gp which is 1 plus x plus x square plus x cube plus and so on Mathematical geometric series the corresponding terms of the a p and gp. Now the series is trig series as the given trig series is given by minus 1 the whole into 3 and this whole multiplied by x to the power n minus 1 and this whole into x to the power n minus 1. This is the nth term of the given mathematical geometric series. Now we suppose we equal to plus and so on plus for the geometric series the multiplying both sides for x we get x into sn is equal to we equal to and this whole into x to the power n minus 1. So here we x to the power n minus 1 like by x give us minus 1 the whole into x to the power n minus x. The whole into sn is equal to 2 square gives us 3x minus 1 into x to the power n minus 1 x to the power 3n minus 1 minus and this whole into x to the power n minus 1 which is equal to 3 into x to the power n minus 1 power n minus 1 and this minus 1 the whole into x to the power n. Now as you can see this is the geometric series. Now we will find out the sum of this geometric series x cube plus and so on 2x to the power n minus 1 of this geometric series is equal to the first term which is 3x into 1 minus the ratio that is x for x to the power minus 1 terms. So this means we have 1 minus x the whole into sn is equal to 2 plus 1 minus x to the power n minus 1 and this whole upon minus 1 whole into x to the power n equal to 2 upon 1 minus x 1 minus x to the power n minus 1 and this whole upon 1 minus x square minus 1 whole into x to the power n and this whole upon 1 minus x. This is the sum of the series to n terms as we can say that the series is equal to 2 upon 1 minus x x into 1 minus x to the power n minus 1 and this whole upon 1 minus x to the power n and this whole upon 1 minus x. Sure x is less than 1 to infinity. This infinity is the sum of the series limit minus 1 the whole and this whole upon 1 minus x to the power square minus 1 this whole in 1 minus x. Now as limit n tends to infinity and x is less than 1 so this x to the power n minus 1 would be 0 or n would also be 0. This is equal to 2 upon 1 minus 0 the whole upon 1 minus x to the power square minus x into 0 or you can say this is equal to 2 upon 1 minus x to the power 1 minus x infinity equal to 2 upon sum of the series to infinity. This is the answer for sum of the series to n terms. Hope you have understood the solution of this question.