Question

Let the coefficients of third, fourth and fifth terms in the expansion of $\left( x +\frac{ a }{ x ^{2}}\right)^{ n }, x \neq 0$, be in the ratio $12: 8: 3 .$ Then the term independent of $x$ in the expansion, is equal to ______.

Answer 1

$T _{ r +1}={ }^{ n } C _{ r }( x )^{ n - r }\left(\frac{ a }{ x ^{2}}\right)^{ r }$
$={ }^{ n } C _{ r } a ^{ T } x ^{ n -3 r } $
${ }^{ n } C _{2} a ^{2}:{ }^{ n } C _{3} a ^{3}:{ }^{ n } C _{4} a ^{4}=12: 8: 3$
After solving
$n =6, a =\frac{1}{2}$
For term independent of 'x' $\Rightarrow n =3 r$
$r =2$
$\therefore $ Coefficient is ${ }^{6} C _{2}\left(\frac{1}{2}\right)^{2}=\frac{15}{4}$
Nearest integer is $4 .$