Which term is independent of x, in the expansion of

Question:

Which term is independent of $x$, in the expansion of $\left(x-\frac{1}{3 x^{2}}\right)^{9} ?$

Solution:

Suppose $T_{\mathrm{r}+1}$ is the term in the given expression that is independent of $x$.

Thus, we have :

$T_{r+1}={ }^{9} C_{r} x^{9-r}\left(\frac{-1}{3 x^{2}}\right)^{r}$

$=(-1)^{r}{ }^{9} C_{r} \frac{1}{3^{r}} x^{9-r-2 r}$

For this term to be independent of $x$, we must have

$9-3 r=0$

$\Rightarrow r=3$

Hence, the required term is the 4 th term.

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