Find a , b and n in the expansion of (a+b)^n if the first three terms in the expansion are 729, 7290, 30375

ncert cbse chapter 8 binomial theorem miscellaneous exercise

 1.Find a , b and n in the expansion of (a+b)^n if the first three terms in the expansion are 729, 7290, 30375

 

(a+b)^n  = (a^n) + C(n,1)[a^(n-1)][b]++ C(n,2)[a^(n-2)][b^2] +...+(b^n)

using the given data and also 

C(n,1) = n

C(n,2) = {n(n-1)}/{2}

we get 

a^n = 729--------------------(1)

n[a^(n-1)][b] = 7290--------------------(2)

{n(n-1)}/{2}*[a^(n-2)][b^2] = 30375---------------(3)

(2)/(1) gives {nb} / {a} = 10 ----------------------(4)

(3)/(2) gives {(n-1)b}/{2a} = {25} / {6}-----------(5)

 

(5)/(4)  gives {n-1}/{2n} = {5}/{12}

therefore 12(n-1)=5(2n)

12n-12 =10n

12n-10n =12

2n=12

n=6 

use in (1)

a^6 = 729

a = 3

use these values of n and a in (4)

{6*b}/{3} = 10

gives b =5

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