 Hello and welcome to the session. I am Asha and I am going to help you with the following question that says, find A, B and N in the expansion of A plus B raised to the power N if the first three terms of the expansion are 729, 7290 and 30,375 respectively. Let us now begin with the solution and we are given that the first three terms are 729, 7290 and 30,375. So let T1 be the first term of expansion A plus B raised to the power N. So this is equal to A raised to the power N which is 729. Let this be equation number one. Now second term T2 of expansion A plus B raised to the power N is equal to Nc1 to A raised to the power N minus 1 into B and this is 729,0. Let this be equation number two. And the third term of the expansion is Nc2 A raised to the power N minus 2 into B raised to the power 2 and this is equal to 30,375. So let this be equation number three. Now dividing equation 1 by 2 that is T1 upon 32 will be equal to 729 upon 729,0. That is A raised to the power N upon Nc1 into A raised to the power N minus 1 into B is equal to 1 upon 10. It is further equal to A upon NB is equal to 1 upon 10. So let this be equation number four and now dividing equation 2 by 3 we get T2 upon 33 is equal to 729,0 upon 30,375 which is further equal to, now T2 is the second term which is Nc1 A raised to the power N minus 1 into B upon and the third term is Nc2 A raised to the power N minus 2 into B square. So this is equal to on simplifying we get 6 upon 25 frame. Now C1 is factorial N upon factorial 1 into factorial N minus 1 E2 is equal to factorial N upon factorial 2 into factorial N minus 2. So this can further be written as N factorial upon 1 factorial into N minus 1 factorial into A raised to the power N into A raised to the power minus 1 into B upon factorial N upon factorial 2 into N minus 2 factorial into A raised to the power N into A raised to the power minus 2 into B square so this is equal to 6 upon 25. So this further implies N factorial cancels up with N and we can further write it as 1 upon N minus 1 into N minus 2 factorial into A raised to the power minus 1 A raised to the power N cancels up with A raised to the power minus A raised to the power N. So we have into B cancels up with B we have N to 1 upon in the denominator we have 1 upon 2 factorial is 2 into N minus 2 factorial A raised to the power minus 2 into B so this is equal to 6 upon 25 or it can further be written as N minus 2 factorial cancels up with N minus 2 and we have 2 upon N minus 1 and A raised to the power minus 2 goes on the numerator and we have A raised to the power 1 upon B. So this is equal to 6 upon 25 or we can say that 2A upon N minus 1 into B is equal to 6 upon 25 so this is equation number 5 and now dividing equation 4 by 5 to get the values of A, B and N so 4 raise A upon N B and we have 2A upon N minus 1 into B so this is equal to 1 upon 10 upon 6 upon 25 or it can further be written as A upon N B into N minus 1 into B upon 2A which is equal to 1 upon 10 into 25 upon 6. So we have 5 2's at N, 5 5's at 25 and A cancels up with A, B with B and we further have N minus 1 upon 2N is equal to 5 upon 12 or on cross multiplying 12 into N minus 1 is equal to 10N or 12N minus 12 is equal to 10N or 12N minus 10N is equal to 12 or 2N is equal to 12 or we can say that N is equal to 12 upon 2 which is equal to 6 so the value of N is 6 on substituting N is equal to 6 and equation number 4 we have A upon N B is equal to 1 upon 10 that is upon 6B is equal to 1 upon 10 or we have A upon B is equal to 6 upon 10 so let this be equation number 6 now substituting A is equal to 6 and equation number 1 which is A raised to the power N is equal to 729 so we have A raised to the power 6 is equal to 729 can be written as 3 raised to the power 6 so this implies A is equal to 3 and now on substituting A is equal to 3 and equation number 6 we have A upon B is equal to 6 upon 10 or 3 upon B is equal to 6 upon 10 so this implies B is equal to 10 into 3 upon 6 so we have 3 2s or 6 2 into 5 is 10 so this implies B is equal to 5 A is equal to 3 equal to 5 and N is equal to 6 so this completes the session take care and bye for now