 Hi and welcome to the session. I am Asha and I am going to help you with the following question that says find n if the ratio of the fifth term from the beginning to the fifth term from the end in the expansion of 4th root of 2 plus 1 upon 4th root of 3 raised to the power n is root over 6 is to 1 This is now beginning with the solution and the fifth term in the expansion of 4th root of 2 plus 1 upon 4th root of 3 raised to the power n is given by nc4 into 4th root of 2 raised to the power n minus 4 into 1 upon 4th root of 3 raised to the power 4 It is further equal to nc4 2 raised to the power n minus 4 upon 4 into 1 upon 3 raised to the power 4 upon 4, which is further equal to n c4 2 raised to the power n minus 4 upon 4 into 1 upon 3 of the total number of terms is equal to n plus 1 the end equal to plus 1 minus 5 plus 1 at term from the beginning That is the term from the beginning very equal to n c is 4 both root of 2 raised to the power n minus n minus 4 into 1 upon 4th root of 3 raised to the power n minus 4 So this is further equal to 3 n minus 4 is equal to factorial n upon factorial n minus 4 into factorial 4 and Similarly nc4 is also equal to factorial n upon factorial 4 into factorial n minus 4 so Both these values are equal so on replacing nc n minus 4 by nc4 This is taken further written as mc4 4th root of 2 raised to the power 4 into 1 upon 4th root of 3 raised to the power n minus 4 Equal to t4 Into 2 raised to the power 1 upon 4 into 4 into 1 upon 3 n minus 4 upon 4 which is further equal to nc4 Into 2 into 1 upon 3 n minus 4 upon 4 According to the question we are given that ratio of fifth term From the beginning upon fifth term and the end is equal to t4 upon 4 into 1 upon 3 upon nc4 into 2 into 1 upon 3 raised to the power n minus 4 upon 4 Now this ratio is given as root over 6 upon 1 so this is equal to mc4 cancels out with mc4 and We have 2 raised to the power n minus 4 upon 4 into 1 upon 3 upon 2 into 1 upon 3 raised to the power n minus 4 upon 4 This further implies that root over 6 upon 1 is equal to 2 raised to the power n upon 4 minus 2 on 1 upon 3 raised to the power n upon 4 minus 2 and on cross multiplying we get root over 6 into 1 upon 3 raised to the power n upon 4 minus 2 is equal to 2 raised to the power n minus 4 minus 2 or it can further be written as 6 raised to the power half is equal to 2 upon 1 upon 3 raised to the power n upon 4 minus 2 Which implies that 6 raised to the power half is equal to 6 raised to the power n upon 4 minus 2 And I'm comparing we find that half is equal to n upon 4 minus 2 n upon 4 is equal to half Plus 2 which is equal to 5 upon 2 Or we can say that n is equal to 5 into 4 upon 2 2 to the power 4 is equal to 10 Hence our answer is the value of n is equal to 10 so this completes the session take care and have a good day