 Hi and welcome to the session. I am Asha and I am going to help you with the following question which says between 1 and 31, N numbers have been inserted in such a way that the resultant sequence is an AP. At ratio of 7 and N minus 1 at term s 5 is to 9, find the value of N. Let us now start with the solution and let A1, A2 up to aim N numbers which are inserted between 1 and 31 such that the resulting sequence is an AP. So the AP found will be, first number is 1 and we have A1, A2 up to aim and the last number is 31. So the total number of terms of this AP are N plus 2. Let us rename this sequence, let 1 be the first element this is noted by T1, this is T2, T3 and so on up to TN plus 1 and this is TN plus 2. So there are in all N plus 2 terms. Now the last term which is T, N plus 2 is given by T1 which is the first term plus N plus 2 minus 1 into T and the last term is 31. So here 31 is equal to 1 plus N plus 1 into D or 30 is equal to N plus 1 into D. So this implies T is equal to 30 upon N plus 1. Now we are given that the ratio of 7 and N minus 1 at term is 5 is to 9. So first let us find the 7th term of this AP which is denoted by T8. So TN will be equal to T1 plus 801 into D that is 1 plus 7 into 30 upon N plus 1 which is further equal to M plus 1 plus 210 upon N plus 1. So this is equal to M plus 211 upon N plus 1. So this is the T8 term of the sequence that is M plus 211 upon M plus 1 which is the seventh number inserted between 1 and 31. Now we have to find the M minus 1 at number. So we will have to find TN which will be the M minus 1 at number which is inserted between 1 and 31. So this is equal to T1 plus M minus 1 into D. So T1 is 1 plus M minus 1 and D is 30 upon M plus 1. So this is equal to M plus 1 plus 30M minus 30 upon M plus 1. So this is equal to 31M minus 29 upon M plus 1. Therefore a M minus 1 term or we can say that TMS term is 31M minus 29 upon M plus 1. Now according to the question we are given that a 7 upon a M minus 1 is equal to 5 is to 9. So this implies that M plus 211 upon M plus 1 which is the seventh term and M minus 1 term is 31M minus 29 upon M plus 1 is equal to 5 is to 9. All we can say that M plus 211 upon 31M minus 29 is equal to 5 upon 9. Now in cross multiplying we have 9 into M plus 211 is equal to 5 into 31M minus 29 or 9M minus 211 upon multiplying 9 to 211 we get 899 is equal to 5 into 31 we get 155 into M minus 1. So multiplying 29 with 5 be it minus 145 or on rearranging we have 9M minus 155M is equal to minus 1899 minus 145 or minus 146M is equal to minus 2044 or M is equal to minus 2044 upon minus 146 is equal to 14. So our answer is the value of M is 14. So this completes the session. Take care and have a good day.