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

Answer

N/a

Explanation

`(a+b)^(n)=^(n)C_(0).a^(n)+^(n)C_(1).a^(n-1).b`
`+^(n)C_(2).a^(n-2).b^(2)+......+^(n)C_(n).b^(n)`
`"Now "" ".^(n)C_(0).a^(n)=729 " "......(1)`
`rArr " "a^(n)=729`
`.^(n)C_(1).a^(n-1).b=7290`
`rArr n.a^(n-1).b^(2)=7290 " "......(2)`
`" and "" "n(n-1) a^(n-2).b^(2) =60750" "......(3)`
Dividing equation (1) by eauation (2)
`(a^(n))/(n.a^(n-1).b)=(729)/(7290)`
`rArr " "(a)/(b) =(1)/(10).n " "......(4)`
Dividing equation (2) by equation (3)
`(n.a^(n-1).b)/(n(n-1)a^(n-2).b^(2)) =(7290)/(60750)`
`rArr " " (a)/(b) =(3)/(25)(n-1) " "......(5)`
From equations (4) and (5)
`(1)/(10)n (3(n-1))/(25)`
`rArr " "30n- 30 =25n rArr n=6`
` a^(6) =729 =3^(6)`
`rArr " " a=3`
From eauation (4)
`(3)/(b) =(1)/(10)xx6`
`rArr " "b=5`
`:. " "a=3,b=5 " and " n=6`