Find a if the coefficients of `x^2`and `x^3`in the expansion of `(3+a x)^9`are equal.

Explanation

We know that

General term of expansion `(a+b)^n` is

` " " " " T_(r+1)= ^nC_r(a)^(n-r).b^r`

For `(3+a x)^9`,

Putting`a=3, b=ax b& n=9`

General term of `(3+a x)^9` is

` " " " " T_(r+1)= ^9C_r(3)^(9-r).(ax)^r`

` " " " " T_(r+1)= ^9C_r(3)^(9-r).a^r.x^r " " " " ....(1)`


We need coefficient of `x^2 & x^3`


Finding coefficient of `x^2`

Putting `r=2` in (1)

` " " " " T_(2+1)= ^9C_2(3)^(9-2).a^2.x^2`

` " " " " T_(3)= ^9C_2(3)^(7).a^2.x^2`

Coefficient of `x^2=^9C_2(3)^(7).a^2`


Finding coefficient of `x^3`

Putting `r=3` in (1)

` " " " " T_(3+1)= ^9C_3(3)^(9-3).a^3.x^3`

` " " " " T_(4)= ^9C_3(3)^(6).a^3.x^3`

Coefficient of `x^3=^9C_3(3)^(6).a^3`


Given that

` " " " " `Coefficient of `x^2`=Coefficient of `x^3`

` " " " " ` `^9C_2(3)^(7).a^2=^9C_3(3)^(6).a^3`

` " " " " ``(9!)/(2!(9-2)!)3^7.a^2=(9!)/(3!(9-3)!)3^6.a^3`

` " " " " ``(9!)/(2!(7)!)3^7.a^2=(9!)/(3!(6)!)3^6.a^3`

` " " " " ``((9!)/(2!(7)!).3^7)/((9!)/(3!(6)!).3^6)=a^3/a^2`

` " " " " ``(9!)/(2!(7)!).3^7xx(3!6!)/(9!.3^6)=a`

` " " " " ``(9!3!6!)/(9!2!7!).(3^7)/(3^6)=a`

` " " " " ``(3xx2!xx6!)/(2!xx7xx6!).3=a`

` " " " " ``(3)/(7)xx3=a`

` " " " " ``(9)/(7)=a`

Hence, `a=(9)/(7)`