Question

Find the term independent of x in the expansion of $(x^2+\frac{1}{x}{)}^9$

Answer 1

We have $T_{r+1}={n_C}_r{x}^{n-r}{a}^r$ where $n=9$

$T_{r+1}={9_C}_r{\left(x^2\right)}^{9-r}{\left(\frac{1}{x}\right)}^r$

$={9_C}_r{x}^{18-2r}{x}^{-r}$

$={9_C}_r{x}^{18-2r-r}$

$={9_C}_r{x}^{18-3r}$

Independent term $\Rightarrow 18-3r=0$

$\Rightarrow 3r=18$

$\Rightarrow r=\frac{18}{3}=6$

$\therefore$ the independent term$=T_7={9_C}_6=\frac{9!}{6!3!}=\frac{9\times 8\times 7\times 6!}{6!\times 3\times 2\times 1}=\frac{9\times 8\times 7}{3\times 2}=3\times 4\times 7=84$

$\therefore$ the term independent of $x$ is $T_7=84$